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BIBLIOGRAPHIC RECORD TARGET 

Graduate Library 
University of Michigan 

Preservation Office 

Storage Number: 



ACQ7946 

UL FMT B RT a BL m T/C DT 07/18/88 R/DT 07/18/88 CC STATmmE/Ll 

010: : I a 05028039 

035/1: : ] a (RLIN)MIUG86-B28757 

035/2: : j a (CaOTULAS)160537099 

040: : | a MnU | c MnU | d MiU 

050/1:0: |aQA371 |b.F7 

100:1 : I a Forsyth, Andrew Russell, | d 1858-1942. 

245:00: | a Theory of differential equations. | c By Andrew Russell Forsyth. 

260: : | a Cambridge, [ b University Press, jcl890-1906. 

300/1: : la4pts. in6v. |c23cm. 

505/1:1 : | a pt I. (Vol. I) Exact equations and Pfaff s problem. 1890.-pt. 

11. (Vol. II-lII) Ordinary equations, not linear. 1900.-pt. III. (Vol. IV) 

Ordinary equations. 1902.--pt. IV (vol. V-VI) Partial differential equations. 

1906. 

590/2: : |aastr: Pt.l (vol 1) only 

650/1:0: | a Differential equations 

998: : | c DPJ | s 9124 



Scanned by Imagenes Digitales 
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THEORY 



DIFFERENTIAL EQUATIONS. 



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aouOon; 0. J. CLAY and SONS, 
CAMBKIDGE UNIVERSITY PEESS WAREHOUSE, 
LANE, 
NGTON STREET, 




m: F. A. EROCKHAUS. 
TUE MAOMILLAN COMPANY. 
ciilfa: MACMILLAN AND 00., Lru. 



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THEOEY 

OF 

DIFFEEENTIAL EQUATIONS. 

PAUT III. 
OKDINAKY LINEAR EQOATIONS. 



ANDREW EUSSELL FOBSYTH, 

ScD., LL.D., F.K.S., 



CAMBRIDGE: 

AT THE UNIVERSITY PRESS. 

1902 

All rights r^ssrued. 



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PBEPACE. 

The present volume, constituting Part III of this 
work, deals with the theory of ordinary linear differential 
equations. The whole range of that theory is too vast to 
be covered by a single volume ; and it contains several 
distinct regions that have no organic relation with one 
another. Accordingly, I have limited the discussion 
to the single region specially occupied by applications 
of the theory of functions ; in imposing this limitation, 
my wish has been to secure a uniform presentation of 
the subject. 

As a natural consequence, much is omitted that 
would have been included, had my decision permitted 
the devotion of greater space to the subject. Thus the 
formal theory, in its various shapes, is not expounded, 
save as to a few topics that arise incidentally in the 
functional theory. The association with homogeneous 
forms is indicated only slightly. The discussion of com- 
binations of the coefficients, which are invariautive under 
all transformations that leave the equation linear, of the 
associated equations that are covaviantive under these 
transformations, and of the significance of these invariants 



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and covariants, is completely omitted. Nor is any appli- 
cation of the theory of groups, save in a single functional 
investigation, given here. The student, who wishes to 
consider these subjects, and others that have been passed 
by, will find them in Schlesinger's Handhuch der Theorie 
der linearen Differentialgleichungen, in treatises such as 
Pieard's Corns d'Analyse, and in many of the memoirs 
quoted in the present volume. 

In preparing the volume, I have derived assistance 
from the two works just mentioned, as well as from the 
uncompleted work by the late Dr Thomas Craig. But, 
as will be seen from the references in the text, my main 
assistance has been drawn from the numerous memoirs 
contributed to learned journals by various pioneers in the 
gradual development of the subject. 

Within the limitations that have been imposed, it 
will be seen that much the greater part of the volume is 
assigned to the theory of equations which have uniform 
coefficients. When coefficients are not uniform, the 
difficulties in the discussion are grave : the principal 
characteristics of the integrals of such an equation have, 
as yet, received only slight elucidation. On this score, 
it will be sufficient to mention equations having algebraic 
coefficients : nearly all the characteristic results that have 
been obtained are of the nature of existence-theorems, 
and little progress in the difficult task of constructing 
explicit results has been made. 

Moreover, I have dealt mainly witli the general 
theory and have abstained from developing detailed 
properties of the functions defined by important par- 
ticular equations. The latter have been used as illustra- 
tions ; had they been developed in fuller detail than is 



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given, the investigations would soon have merged into 
discussions of the properties of special ■ functions. In- 
stances of such transition are provided in the functions, 
defined by the hypergeometric equation and by the 
modern form of Lamp's equation respectively. 



A brief summary of the contents will indicate the 
actual range of the volume, In the first Chapter, the 
synectic integrals of a linear equation, and the conditions 
of their uniqueness, are investigated. The second Chapter 
discusses the general character of a complete system of 
integrals near a singularity of the equation. Chapters 
III, IV, and V are concerned with equations, which have 
their integrals of the type called regular ; in particular. 
Chapter V contains those equations the integrals of which 
are algebraic functions of the variable. In Chapter VI, 
equations are considered which have only some of their 
integrals of the I'egular type ; the influence of such 
integrals upon the reducibility of their equation is in- 
dicated. Chapter VII is occupied with the determination 
of integrals which, whUe not regular, are irregular of 
specified types called normal and subnormal ; the 
functional significance of such integrals is established, 
in connection with Poincare's development of Laplace's 
solution in the form of a definite integral. Chapter VIII 
is devoted to equations, the integrals of which do not 
belong to any of the preceding types ; the method of 
converging infinite determinants is used to obtain the 
complete solution for any such equation. Chapter IX 
relates to those equations, the coefficients of which are 
uniform periodic functions of the variable : there are two 



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classes, according as the periodicity is simple or double. 
The final Chapter deals with equations having algebraic 
coefficients; it contains a brief genera! sketch of Poincare's 
association of such equations with automorphic functions. 



In the revision of the proof-sheets, 1 have received 
valuable assistance from three of my friends and former 
pupils, Mr. E. T. Whittaker,M.A., and Mr. E. W. Barnes, 
M.A., Fellows of Trinity College, Cambridge, and Mr, 
E. W. H. T. Hudson, M.A., Fellow of St John's College, 
Cambridge ; I grateftilly acknowledge the help which 
they have given me. 

And I cannot omit the expression of my thanks to the 
Staff of the University Press, for the unfailing courtesy 
and readiness with which they have lightened my task 
during the printing of the volume. 

A. R. FORSYTH. 



Trisity Cult.bge, Cambeidgb, 
1 jtfareA, 1902. 



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CONTENTS. 



CHAPTER I. 

LINEAR EQUATIONS : EXISTENCE OF SYNECTIC INTEGRALS ; 
FUNDAMENTAL SYSTEMS. 

Introductory remarks 

Form of the homogeneous linear equation of oidei m 

Eatablisbmeut of the esistence of a sjneutii, integrril in the 
domeiin of an ordinary puint, dotennined uniquely by the 
initial conditions with corollanea, and Bsamples 

Hermite's treatment of the equation with constant tooflicienta 

Continuation of the aynectie integral beyond the initial 
domain ; r^on of its continuity bounded by the singu- 
larities of the equation 

Certain deformations of path of independent variable leave 
the final int^ral unchanged ...... 

Seta of integrals determined by sets of initial valuea . 

The determinant ^ {z) as affecting the linear independence 
of a sot of m integials a fundamental system and the 
eSective test of it* fitness 

Any integial is linearly e:£press!ble in terms of the elements 
of a fundlmentul sjstem ..... 

Construction if i speiiil fundamental systeni 



CHAPTER II. 
GENERAL FORH AND PROPERTIES OP INTEGRALS NEAR A SINGULARITY. 

13. Constructitn of the fundwmentcd equation belonging to a 

singularity 35 

14. The fundimental equation is independent of the choice of the 

f\indimcntal aytera Poincarii'a theorem ... 38 



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CONTENTS 



Tl 1 m tay d &ors f tt f d t ! q 

it la tal f m L n 

T y 13 p joat w b 11 

4. fi d m ta] y t f t^r 1 1: th 

f dm tal ©q tat diat t f 

Eff t f m Itpl oot 
T fmt fthfdmtleqt 

1 f th d ced f n 

l_ 1 f teiral ted with Itil 

fth g p t bgi [ 
TI bmg b g 1 eq t OB h t 

tegral t g" P d th g 

i t th fegnl 

■y I hoat f th Ijt 1 I f th 

Hfti b b-g p 

ult th teg al 



1 



IM 



f 1 



relating to such integrals . 



f tl 
lyt I 



CHAPTER III. 

RRQULAR ISTEGBALS 1 EQUATION HAVING ALL ITS INTEGRALS RBRTJI.AI 
NEAR A SINGULARITY. 

39. Definition of integral, regular in the vicinity of a singularity : 

index to which it belongs 7 

30. Iddes of the dctemiioant of a fundamental system of integrals 

all of which are regular near the singularity ... 7 

31. Form of homogeneous linear equation when all its integrals 

are regular near a 7 

32. 33. Converse of the preceding reault, established by the method 

of Frobenius ......... 7 

34. Series proved to converge uniformly ajid unconditionally . 8 

35, Integral associated with a simple root of an algebraic (in- 

dicia!) equation ..,.■... i 
36—38. Sot of int^rals associated with special group of roots of the 
algebraic (indicial) equation, with summaj'y of results 
when all the integrals are regular E 

39. Definition of indieial equation, indidol funetiiyii : signiiicance 

of integrals obtained i 

40. The iiit^rals obtained constitute a fundamental system ; with 



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Conditiona that ever3 iPgular integral belonging to a par- 
ticulir exiionent '.hould have its espression free from 
logiiithms , with es^mple'J ...... 

Condititni that there should be at least one r^ular integral 
belonging to a partitulai exponent and free from loga- 
iithms ...... 

Alternitue method aometimea effective for settling the ques- 
tion in §§ 43, 43 

Discnmination between real singularity and apparent singu- 
lant; conditions fot the latter ..... 

Noh on thp leties in 1) S4 



CHAPTER IV, 

EQUATIONS HAVING THEIR INTEGKALS REGULAR IN THE VICINITY OF 
EVERY SINGULARITY (iKCLOUING infinity). 

46. Equations (said to be of the FacAsian type) having all their 

int^r&ls r^ular in the vicinity of every singularity 
{including =o ) ; their form : with examples . . . 123 

47. Equation of second order completely determined by assign- 

ment of singularities and tlieir exponents : Riemaun's 
jp-function 135 

48. Significance of the relation among the exponents of the pre- 

ceding equation and fimotion 139 

49. Oonatruction of the differential equations thus determined . 141 

50. The equation satisfied by the hype^eometric series, with 

51. 52. Equations of the Fuclisian type and the second order with 

more than three singularities (i) when co is not a singu- 
larity, (ii) when oo is a singularity ..... 150 

53. Normal forme of such equations 156 

54. Lamp's equation transformed so as to be of Fuchsiaa type . 160 

55. ■ B6clier'a theorem on the relation between the linear equations 

of mathematical physics and an equation of the second 
order and Fuchaian type with five singularities . . 161 

56. Heine's equations of the second order having an integral that 

is a polynomial 165 

57. Equations of the second order all whose integrals are rational 169 



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CHAPTER V. 

1 KQUATIONS OP THE SECOND AKD THE THIRD OEDERS 
i ALGEBRAIC I 



Mutlnj la of letermining whether an equation hi^ dlt,pl i ii 

ntegi ill 
Kle a a special meth id for detonnining all the finite groups fc 

thp equation of the ^uond ordei 
Ihp pq ationa satisfied by the quotient of two 'olution-j 

the equition of the second order then integrals 
L m'iti-urtion of equations of the second order a" 

ntegrable 
Meini of deteiminmg whether a gi\eii equation is n 

braicaily uitegralle with examples 
Lqu-itims jf the third order their quotient eq lat ons 
Painleve >■ invariants, cirreaponding to the Scliwaiziin der 

tive for the equation of the second order connet 

with La^uerres mvanant 
A'.i loiation with finite group? of trai aformat oi 

1 neo-linear in two variables 
IndiLitims Df other poMible methods 
Eemiik-J on equatmn^ of the foiuth. orler 
Association of equations of the third ani highei triers 

the theon of homogeneous forms 
And of equati >ns of the second order 
"Diacuasion of equations of the thiri order with i -"e 

theorem due to Fuch-i with cs'implp in I roferei < 

e j^uations of higher order 



th^t ' 



CHAPTER VI. 

lUA'l'IONS HAVING ONLY SOME OF THEIR INTEGHALS REGULAR 
NEAR A SINGULARITY. 

Equations having only some of their integrals regular ii 
vicinity of a singularity : the characteristic indes . 

The linearly independent aggregate of r^ular integrals 
satisfy a liiieaj equation of order equal to their number , 

Reducible equations 

Frobenius's characteristic function, indicial function, ivdiciid 
eqwitwji ; normal form of a differential equation associ 
able with the indicia] function, uniquely determined b; 
the characteristic function 

The number of r^ular iutegrals of an equation of order ii 
and oharaoterietic index n is not greater than m — n 



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CONTENTS 



Tlr mbe f gul tp ! be 1 sa th 

D te t f tl regul t 1 h tii y t 

w th mpl 
E t f irred bl equat 

Aqt fil h g dpedt ulor te 

b las grl t 1 a. tedw th 

it f rd 
La eqtndf to^n qt 

Eelt htee qt dtadit, pett f 

th L fheil Ipdtrel t orals 

Jia dhlt I t 



CHAPTER Vn. 



NORMAL integrals: suenosmal integrals. 

Integrals for which the singularity of the equation is 

essential : normal int^rals 

Thome's method of obtaining norma! integrals when they 

Construction of determining factor : possible cases . 
Svhnormal integrals 

Rank of an equation ; Poincar^'s theorem od a set of 
normal or subnormal functions as integrals ; examples 

Hftmbu:^er's equations, having s=0 for an essential singu- 
larity of the integrale, which are regular at tc and 
elsewhere are sjnectic : equation of second order 

Cayley's method of obtaining normal iut^rals . 

Hamburger's equations of order higher than the second 

Conditions associated with a simple root of the character- 
istic equation for the determining factor . 

Likewise for a multiple root 

Subnormal int^rals of Hamburger's equations 

Detailed discussion of equation of the third order . 

Normal integrals of equations with rational coefficients 

Poincar^s development of Laplace's solution for grad( 

1. Liapouuoff's theorem ....... 

1—105. Application to the evaluation of the definite int^ral ir 
Laplace's solution, leading to a normal integral 

i. Double-loop integrals, after Jordan and Pochhammer 

'. When the normal series diverges, it is an asymptotic repre- 

sentation of the definite-integral solution 

i. PoincarS's transformation of equations of rank higher than 

unity to equations of rank unity 



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CHAPTER V1.IT. 

INFINITE DETERMISANTS, AND THEIK APPLICATION TO THE SOLUTION 
OF LINBAU EQUATIONS. 

109. lutljluU 311 Ji iniimte determnints tLsts of c l\OY- 

s;cnci, 1 io[>eities . 348 

110. Dciebitiicnt . 352 

111. Mm I'l 353 

112. IT[ ftrm comergence when t iistitui.nt-< aie fun t ons of ii. 

paiametei . 358 

113. Solution of an unlimited r umbct of simultaneous In ear 

equations . 360 

114. Difletential equations having no regular totegial oo noimal 

integial no subnormal integtal 363 

115. Inf-egral in the form (f a Laurent aenes intioducticn of 

!iii inhmte determinant ii{pj ■ SGii 

IIG. C nvergent* ot Q(p) . 366 

117. Introiuction of i^nother inimite deteiminint I>{p) its 

convergente and its telation to Q{p) with deduced 
eiiieSBion of £2(p) . 369 

118. ( niei^ence of tlie Liurent ^nea exj reding the mtegial . 376 

119. & neiihsation of mpthod ol Fiobennis (in Ohap Hi) to 

determine a system cf mtBgrah . 379 

120 — 123. 'V annua cases atoordmg t) the chaiatter of the ineduLiblc 

r ots of i>{p) = . 380 

124. The i^stem ot integrals i5 fundamental . 387 

126. Th equati n l>{p)=0 is efiectneh the fundaineatal equa- 

tion for the combiiidtnn of Bingiilantiei within the 
cu'cle I |=fi . 389 

126. (.eneiil remark example'. . 392 

127. Other methc h of ibtaining the fundamental equati ii to 

which D (fi)=0 IS efiectively equivalent with an 
example . 398 



CHAPTER IX. 

EQUATIONS WITH UNIFOHII PLRIODIC f DLIf ICIENTS. 

Equation-fl with mmply 'penodu, coefhcients the funda- 
mental er[uation a sooiated with the juniod 
Siiiple looti f the fund^mentil equat on 
\. iQultii le 1 not of the fundamental equiti ti . 



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131. Analytical form of tte integrals associated with a root . 411 

132. Modification of the form of the group of int^rals associated 

with a multiple root 414 

133. Use of elementary divisors : resolution of group into sub- 

groups : numbei' of integrals, that are periodic of the 
second kind 416 

134. More precise establiahment of results in § 133 . . . 417 

135. Converse proposition, analogous to Fuchs's theorem in § 25 420 

136. Further determination of the integrals, with examples . 421 

137. Liapounoflfs method 425 

138 — 140. Discussion of the equation of the elhptic cylinder, 

v/' + {a-i-ocOB^)w=^0 . . . .431 

141. Equations with doubly-periodic coefficients; the funda- 

mental equations associated with the periods . . 441 

143, 143. Picard's theorem that such an equation poasessea an 

int^ral which is doubly- periodic of the second kind : 

the number of such integrals 447 

144, 145. The int^rals associated with multiple roots of the funda- 

mental equations ; two cases ..... 451 

146. First stage in the construction of analytical expressions of 

integrals ......... 457 

147. Equations that have uniform integrals : with examples . 459 

148. Lamp's equation, in the form vf=w{n{n + \)i^{z)-^B), 

deduced from the equation for the potential . . 464 
149 — 151. Two modes of constructing the integral of Lame's equation 4li8 



CHAPTER X. 

EQUATIONS HAVING ALGEBRAIC COEFFICIENTS. 

152. Equations with algebraic coefficients 478 

153, 154. Fundamental equation for a singularity, and fundamental 

systems; examples 480 

155, 156. Introduction of automorphic functions .... 498 

167, 158. Automorphic property and oonformal representation . 491 

159—161. Automorphic property and linear equations of second order 495 

162. Illustration from elliptic functions 501 

163. Equations with one singularity 506 

184. Equations with two singularities 507 

165. Equations with three singularities 508 

166. General statement as to equations with any number of 

singularities, whether real or complex . . . BIO 



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statement of Poiucar^'s results 

Poincar^'s theorem that any linear equation with algebraic 
coefficients can be integrated by Fucbsian and Zeta- 
fucheian functions 

Properties of these functions : and verification of Poincar^'s 
theorem 

Concluding remarks 



IN BBS I 



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CHAPTER I. 

Linear Equations ; Existence of Synectic Integrals : 
Fundamental Systems. 

1. The course of the preceding investigations has made it 
manifest that the discussion of the properties of functions, which 
are defined by ordinary differential equations of a general t3rpe, 
rapidly increases in difficulty with successive increase in the order 
of the equations. Indeed, a stage is soon reached where the 
generality of form permits the deduction of no more than the 
simplest properties of the functions. Special forms of equations 
can be subjected to special treatment ; but, when such special 
forms conserve any element of generality, complexity and difficulty 
arise for equations of any but the lowest orders. There is one 
exception to this broad statement ; it is constituted by ordinary 
equations which are hnear in form. They can be treated, if not 
in complete generality, yet with sufficient fulness to justify their 
separate discussion ; and accordingly, the various important results 
relating to the theory of ordinary linear differential equations 
constitute the subject-matter of the present Part of this Treatise. 

Some classes of linear equations have received substantial 
consideration in the construction of the customary practical 
methods used in finding solutions. One particular class is com- 
posed of those equations which have constants as the coefficients 
of the dependent variable and its derivatives. There are, further, 
equations associated with particular names, such as Legendre, 
Bessel, Lam^ ; there are special equations, such as those of the 
hypergeometric series and of the quarter-period in the Jacobian 
theory of elliptic functions. The formal solutions of such equations 



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2 HOMOGENEOUS [1 . 

can be regarded as known; but so long as the investigation is 
restricted to the practical construction of the respective series 
adopted for the solutions, no indication of the range, over which 
the deduced solution is valid, is thereby given. Ifc is the aim of 
the general theory, as applied to such equations, to reconstruct 
the various methods of proceeding to a solution, and to shew 
why the isolated rules, that seem so sourceless in practice, actually 
prove effective. In prosecuting this aim, it will be necessary to 
revise for linear equations all the customarily accepted results, so 
as to indicate their foundation, their range of validity, and their 
signiiicance. 

For the most part, the equations considered will be kept as 
general as possible within the character assigned to them. But 
from titne to time, equations will be discussed, the functions 
defined by which can be expressed in terms of functions already 
known ; such instances, however, being used chiefly as illustrations. 
For all equations, it will be necessary to consider the same set of 
problems as present themselves for consideration in the discussion 
of unrestricted ordinary equations of the lowest orders : the exist- 
ence of an integral, its uniqueness as determined by assigned 
conditions, its range of existence, its singularities (as regards 
position and nature), its behaviour in the vicinity of any singu- 
larity, and so on : together with the converse investigation of the 
limitations to be imposed upon the form of the equation in order to 
secure that functions of specified classes or types may be solutions. 
As is usual in discussions of this kind, the variables and the 
parameters will he assumed to be complex. It is true that, for 
many of the simpler applications to mechanics and physics, the 
variables and the parametei'S are purely real ; but this is not the 
case with all such applications, and instances occur in which the 
characteristic equations possess imaginary or complex parametei-s 
or variables. Quite independently of thk latter fact, however, it 
is desirable to use complex variables in order to exhibit the proper 
i-elation of functional variation. 

2. Let z denote the independent variable, and w the dependent 
variable ; z and w varying each in its own plane. The differential 
equation is considered linear, when it contains no term of order 
higher than the first in w and its derivatives ; and a linear equation 
is called homogeneous, when it contains no term independent of w 



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2.] LINEAR EQUATIONS 3 

and its derivatives. By a well-known formal result*, the solution 
of an equation that is not homogeneous can be deduced, merely by 
quadratures, from the solution of the equation rendered homo- 
geneous by the omission of the term independent of w and its 
derivatives ; .and therefore it is sufficient, for the purposes of the 
general investigation, to discuss homogeneous linear equations. 
The coefficients may be uniform functions of s. either rational or 
transcendental ; or they may be multiform functions of a, the 
simplest instance being that in which they are of a form (s, z), 
where is rational in s and z, and s is an algebraic function of z. 
Examples of each of these classes will be considered in turn. The 
coefficients will have singularities and (it may be) critical points ; 
all of these are determinable for a given equation by inspection, 
being fixed points which are not affected by any constants that 
may arise in the integration. Such points will be found to include 
all the singularities and the critical points of the integrals of the 
equation ; in consequence, they are frequently called the singu- 
larities of the equation. Accordingly, the differential equation, 
assumed to be of order m, can be taken in the form 

where the coefficients p^, p^, ..., pm are functions of s. In the 
earlier investigations, and until explicit statement to the contrary 
is made, it will be assumed that these functions of z are uniform 
within the domain considered ; that then' singularities are isolated 
points, so that any finite part of the plane contains only a limited 
number of them : and that all these singularities (if any) for finite 
values of s are poles of the coefficients, so that their only essential 
singularity (if any) must be at infinity. Let f denote any point in 
the plane which is ordinary for all the coefficients p ; and let a 
domain of ^ be constructed by taking all the points z in the 
plane, such that 

|2-fi«i«-fi. 

where a is the nearest to ^ amohg all the singularities of all the 
coefficients. Then within this domain (but not on its boundary) 
we have 

P.-P.i'-t). (»-1.2 -»). 

* See mj Treathte on Differential Eqnatiinui, § 75. 



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4 SYNECTIC [2. 

where P„ denotes a regular function o{ s — ^, which generajly is 
an infinite series of powers of z — f converging within the domain 
of ^. An integral of the equation existing in this domain is 
uniquely settled hy the following theorem ; — 

In the domain of an ordinary point ^, the differential equation 
possesses an integral, which is a regular function of z — ^ and, with 
its first m — 1 derivatives, acquires arbitrarily assigned values when 
z = Z; and this integral is the only regular function of z—^ in 
the specified domain, which satisfies the equation and fulfils the 
assigned conditions*. 

The integral thus obtained will be calledf the synectio integral. 

Synectic Integrals. 

§. The existence of an integral which is a holomorphic 
function of s— ^ within the domain will first be established. 

Let r he the radius of the domain of i^; let M,, ..., M^ denote 
quantities not less than the maximum values of \p,], ..., \pm\ 
respectively, for points within the domain ; and let dominant 



ictions ^, ... 


, <|ini, defined by the expressions 


constructed. 


Then* 







for every positive integer a. The dominant functions ^ are used 
to construct a dominant equation 

^ = ■ft S^S=r + <^ rf^S^ + - + ■^'"«. 
which is considered concurrently with the given equation, 

* The conditions, as to the arbitrarily assigned values to be ftotiuired at f by tu 
and itfl derivatives, are called the initial conditions ; the values are called the 
initial values, 

t As it is a regular function of the variable, it would have been proper to call 
it the regular int^cal. This term has however been appropriated [sec Chapter iii, 
§ 39) to describe another class of integrals of linear equations; as the use in this 
other conneciion is now widespread, oonfusion would result if the use were changed. 

J See mj Theory of Functions, 2aA edn,, §22; quoted hereafter as ?'. J''. 



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3,] INTEGRALS 5 

Any function which is regular in the domain of ^ can be 
expressed as a converging series of powers of ^~f; and the 
coefficients, save as to numerical factors, are the values of the 
various derivatives of the function at f Accordingly, if there is 
an integral w which is a regular function of 5 — J^, it can be formed 
when the values of all the derivatives of w at £f are known. To 

w. -r- , .--, ~, , the arbitrary values specified in the initial 

conditions are assigned. All the succeeding derivatives of w can 
be deduced from the differential equation in the form 



" rf^""" 



I- A^,^', 



(for a — m, m + 1, ... ad inf.), by processes of differentiation, 
addition, and multiplication: as the coefficient of the highest 
derivative of w in the equation (and in every equation deduced 
from it by differentiation) is unity, new critical points are not 
introduced by these processes, so that all the coefficients A are 
regular within the domain of f. 

The successive derivatives of u are similarly expressible in the 

(for a = m, m + 1, ... ad inf.), obtained in the same way as the 
equation for the derivatives of w. The coefficients B have the 
same form as the coefficients A, and can be deduced from them by 
changing the quantities p and their derivatives into the quantities 
tp and their derivatives respectively. 

The values of the derivatives of w and u a,t ^ are required. 
When i^ = J", all the terms in each quantity B are positive ; on 
account of the relation between the derivatives of the quantities p 
and <b, it follows that 

-B«>|^«|, (s = l, ...,m), 

.... , „, , \dw\ I d'^'^w I , 

when 2 = f. Let the mitial values oi |wl, -j- . ■■-, . m^' > when 

z = ^, be assigned as the values of u, f ,■■■• -j^i^i when 2 = ^; 
then 



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6 






EXISTENCE 


OF 






when«- 


r. 


for the valu 


Mm, m + 1, 


... oi 


:». If the 


, series 






(») + (2- 


/A.\ (i 


-0- 








"Ui + 


2! 




converge! 




-"© 


denotes the 


v.,.e„f^'. 


i'hen z 


series 















where ( -r— 1 denotes the value of -^— when z=t, also converses ; 
Vets"/ ffla" ° 

it then represents a regular function oi z~ ^ which, after the mode 
of formation of its coefficients, satisfies the differential equation. 

We therefore proceed to consider the convergence of the series 
for u, obtained as a purely formal solution of the dominant equa- 
tion. To obtain explicit expressions for the various coefficients in 
this series, let s — f = rw, taking x as the new independent variable. 
Points within the domain of f are given by |a;|< 1 ; and the 
dominant equation becomes 

ax™ s=i dx"^' 

When the .series for u, taken in the form 



is substituted in the equation which then becomes an identity, a 
comparLsoQ of the coefficients of a^ on the two sides leads to the 
relation 

holding for all positive integer values of k. 

This relation shews that all the coefficients h are expressible 
linearly and homogeneously in terms of 6o, 6i, ..., hw-i- and that, as 
the first m of these coefficients have been made equal to the moduli 
of the in arbitrary quantities in the initial conditions for w and 
therefore are positive, all the coefficients h are positive. Hence 
k + M,r, 



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S.] A SYNECTIC INTEGRAL 7 

By the initial definition of M,, it was taken to be not less than 
the maximum value of \pi\ within the domain of f; it can there- 
fore be chosen so as to secure that M^r > m. Assuming this 
choice made, we then have 

OjK+l > Om+i;— 1 > 

so that the successive coefficients 



From the difference- equation satisfied by the coefficients b, it 
follows that 

V+*-. k + ni ,^2 (m + k)l ' t™+ft_,' 
So far as regards the m — 1 terms in the summation, the ratio 
^m+ii-s -^ i'm+ft-i is less than unity for each of them ; Mgi^ is finite 
for each of them; and \in + k — s)\-^{m+k)\ is zero for each of 
them, in the limit when k is marie infinite. Hence we have 

and iherefoi'e 



<1. 

for points within the domain of f, so that* the series 

converges within the domain of f. The convergence is not estab- 
lished for the boundary, so that it can be affirmed only for points 
within the domain; it holds for all arbitrary positive values 
assigned to b^, b^, ..., b^-i. 

It therefore follows that, at all points within the domain of ^, 
a regular function of s — i^ exists which satisfies the original 
differential equation for tv, and, with its first m — 1 derivatives, 
acquires at f arbitrarily assigned values. 

4. Now that the existence of a synectic integral is established, 
the explicit expression of the integral in the form of a power-series 
in z — ^, this series being known to converge, can be obtained 

" Cbrystal's Algelra, vol. ii, p. 121. 



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8 UNIQUENESS OF [4. 

directly from tlie equation. As f is an ordinary point for each of 
the coefficients p, we have 

p,^PA.^-0, (s = l, 2, ...,m), 

where Pg denotes a regular function of a — f. Let a^, oli, ..., ow-, 
be the arbitrary values assigned to w, -j— , ..., -j ^~ , when s = f ; 
and take 

which manifestly satisfies the initial conditions. In order that 
this may satisfy the equation, it must make the equation an 
identity when the expression is substituted therein. When the 
substitution is effected, and the coefficients of {e — ^y on the two 
sides of the identity are equated, we have a relation of the form 



where A^+s is a linear homogeneous function of the coefficients a,, 
such that K<m + s, and is also linear in the coefficients in the 
quantities P, (3— f), ..., P„(2 — f); and the relation is valid for 
s= 0, 1, 2, ..., ad inf. Using the relation for these values of s in 
succession, we find a^, a^+i, Om+a- ■■■ expressed (in each instance, 
after substitution of the values of the coeiScients which belong to 
earlier values of s) as a linear homogeneous function of the quanti- 
ties Oj, «!, ..., ctnt-ii and in am+s, the expressions, of which the 
initial constants a^, a^, ..., 0^-1 are coefficients, are polynomials of 
degree s + 1 in the coefficients of the functions P,{s — ^), .... 
Pm (^ - ^)- The earlier investigation shews that the power-series 
for w converges ; accordingly, the determination of the coefficients 
a in this manner leads to the formal expression of an integral w 
satisfying the equation. 

5, Further, the integral thus obtained is the only regular 
function, which is a solution of the equation and satisfies the 
initial conditions associated with a^, eij, ..., 0^-1. If it were 
possible to have any other regular function, which also is a solu- 
tion and satisfies the same initial conditions, its expression would 
be of the form 



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5.] THE STNECTIC INTEGRAL 9 

a regular function oi z — i^. The coefficients would be determin- 
able, as before, fi'om a relation 



whore ^'m+j is the same function of a^, ..., a^-i, «'m. ■■-, <t'ni+a-i 
as ^m+s is of Oo, ..., a^-i, «,„, ..., am+E_i- Hence 

a'jii+i = -^'in+1 = -dm+i> after substitution for a'^, 

and so on, in succession. The coefficients agree, and the two 
series are the same, so that w = w' ', and therefore the initial con- 
ditions uniquely determine an integral of the equation, which is a 
regular function of ^ — f in the domain of the ordinary point ?. 

Corollary I. If all the initial constants ag, a,, ..., a^-i are 
zero, then the synectic integral of the equation is identically zero. 
For in the preceding discussion it has been proved that o-m+e, for 
all the values of s, is a linear homogeneous function of a^, ..., 
Oot-i ; hence, in the circumstances contemplated, a,n+s = for all 
the values of s. Thus every coefficient in the series vanishes ; 
accordingly, the integral is an identical zeio. 

CoEOLLARY II. The initial constants a^, a^, ..., am_i occur 
linearly in the ea^ession of the synectic integral ; and each of the 
m variable quantities, which have those constants /or coe^cients, is 
a synectic integral of the equation. The first part is evident, 
because all the coefficients in w are linear and homogeneous in 
OoiO:,, ..., cW-i. As regards the second part, the variable quantity 
multiplied by Sg is derivable fi'om w by making a^ = 1, and all the 
other constants a equal to zero ; these constitute a particular set 
of initial values which, according to the theorem, determine a 
synectic integral of the equation. Thus the synectic integral, 
determined by the initial values a^, ..., "m-i, is of the form 

aoUi + aiU^+ ... -i-am_iJfm, 
where each of the quantities m,, u^, ...,«„ is a synectic integral of 
the equation. 

j!fote 1. The series of powers oi z — ^, which represents the 
synectic integral, has been proved to converge within the domain 



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10 EXISTENCE OF [5. 

of Z, SO that its radius of convergence is | « - f | , where a is the 
singularity of the coefficients which is nearest to f. All these 
singularities lying in the 6nibe part of the plane are determinable 
by mere inspection of the forms of the coefficients : another 
method must be adopted in order to take account of a possible 
singularity when z = co because, even though a = oo may be aji 
ordinary point of the coefficients, infinite values of the variable 
affect the character of w and its derivatives. 

For this purpose, we may change the variable by the substi- 
tution 

ea:= 1, 
and we then consider the relation of the a'-origin to the trans- 
formed equation as a possible singularity. The transformation of 
the equation is immediately obtained by means of the formula 

#w , ,,t * fc ! (fe - 1) ! a^+° d'w^ 



dz" ^ ^ ,Z^aL\{a.-\)\{k-a)\ dx-' 

inspection of the transformed equation then shews whether x = () 
is, or is not, a singularity. Or, without changing the independent 
variable, we may consider a series for w in descending powers of z : 
pies will occur hereafter. 



It may happen that there is no singularity of the coefficients 
in the finite part of the plane, infinite values then providing the 
only singularity. In that case, we should not take the quantity r 
in the preceding investigation as equal to [co — ^], that is, as 
infinite ; it would suffice that r should be finite, though as large 
as we please. 

It may happen that there is no singularity of the coefficients 
for either finite or infinite values of s; if the coefficients are 
uniform, they then can only be constants. The dominant equa- 
tion is then effectively the same as the original equation ; the 
investigation is still applicable, but it furnishes less information 
as to the result than a method which will be indicated later (§ 6). 

Note 2. The preceding proof is based upon that which is 
given* by Fuchs in his initial, and now classical, memoir on the 
theory of linear differential equations. 

* Creile, t. lkvi (1866), pp. 133—135. 



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5.] A SYNECTIC INTEGRAL 11 

The theorem can also be established by regarding it as a 
particular case of Cauchy's theorem, which relates to the posses- 
sion of unique synectic integrals by a system of simultaneous 
equations. If 

_ ii"w 
™'' ~ ~3^ ' 
the homogeneous linear equation of order 
the system 

-i- = Wj+i, for s = 0, 1, .... 
(is 

These equations possess integrals, expressible as regular functions 
of if—?, such that w„, Wi, ...,w™_, assume arbitrarily assigned 
values when e — ^, and the integrals are unique when thus 
determined : which, in effect, is the theorem as to the syuectic 
integral of the hnear equation*. 

Note 3. A different method for establishing the existence of 
the integrals, though if. does not indicate fully the region of 



(a = 


^0, 1 


»-I), 




1 be 


replaced by 


m-^2, 






-i-pmW< 







their convergence, can be based upon 
Giintherf. It consists in the adoption 


a suggestion 
of another i 


made by 
subsidiary 


equation 










where 






... +-fm'V, 






*'-{^-'^r 






for^=l, ... 
its integrals 


are 


The advantage of this form of equati< 
explicitly given in the form 


an is that 






„.f, -i^fr 







where a- is a root of the equation 

(7 (t - 1) ... (<7 -m + 1) = - rjlf,^ (o- - 1) ... ((7 ^ m + 2) 

+ ^M,a- (<7 - 1) . . . (ff - m + 3) + . . . 
+ {- l)™-'r"'-W™_,.7 + (- l)'"r™M^. 

* See Part ii of this Treatise, gg 4, 10—13. 

I CreUe, t. csviii (1897J, pp. Sol— 3.13 ; see also some remarks thereupon b; 
Fuohs, a.,pp. 354, 353. 



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12 EXAMPLES [5. 

If a root a is multiple, the corresponding group of integrals is 
easily obtained*. 

The construction of the actual proof on the foregoing lines is 
left aa an exercise. 

Ex. 1. Consider the equation 



h arity of the coeffl- 

1 d n P y to the immediate 

po he coefficients of the 

a, idi unity. The equa- 

gr wh h ries of powers of z 

q y d te mm d hy the conditions 

d & wy constant.^. To 



which then must be an identity. In order that the coefficient of 2" may 
vanish after substitution, we must have 

(« + a)(m + I)6„^.5-{«2 + «-^)6„=0, 

Now hy the initial conditions, we have 

h^ = a, 6i = 3; 
hence 

" See my Treatine on Differential Equations, %% 47, 43. 



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5.] EXAMPLES 

and, similarly, 



i products baing taken for integer values of s from 1 to m. The 
syaectic integral satisfying the initial conditions is 

both series, if infinite, converging for values of i such that | ^ | < 1- 

The best known instance of this equation is that whict is usually asso- 
ciated with Legendre's name : k then isp(p + l), ondp (in the simplest form) 
is a positive integer. If p be an even integer, all the coefBcients b^„, for 
2m > p, vanish, so that the quantity multiplying a is then a jwlynomial ; the 
quantity multiplying j3 is an infinite series. If p be an odd integer, all the 
coefBcients fiam + u f"r 2ni.+ l >p, vanish, so that the quantity multiplying 3 
is then a polynomial ; the quantity multiplying a ia an infinite aeries. In all 
other cases, the quantitiea multiplyii^ a and (3 are, each of them, infinite 
aeries ; in every instance, the aeries converge when | z | < 1. 

£V. 2. Obtain the syncctic integral of the equation 

(which includes Bessel'a equation as a special case), with the initial conditions 
that w = a, -T =0 when 2=c, where \o\ > 0. 

Ex. 3. Determine the synectic integral of the equation of the hyper- 
geometric series 

"(l-)^ + {r-(.+»+l).)s-*— », 

the initial conditions being that w = A, -j- =B, when i=^. 

E». 4. Determine the synectic integraJa in the domain of £=0, 
by the equation 

with the initial conditions (i) that w=l, rT-=0, when z=0 ; 
(ii) that w = 0, -£-=h wben 2 = 0. 

£x. 5. Prove that the synectic integral in the domain of 3 = 0, 
by the equation 



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14 EQUATIONS WITH [5. 

with the initial conditions that vi=l, -j- = 0, when ^ = 0, is 

and if the term in w iuvolving s" he — j s", then 

c^, = (,»-2 + (2»-^-« + l)»"-Ha3»-«-(»-4)2-=-^™-i| «-'= + .... 
Prove also that the primitive can he espressed in terms of Bessel's functions 
of order zero and argument — «="' . 

&.-. 6. The equation with constant coefficients may he taken in the form 

which converges everywhere in the finite part of the piano : and «„, ..., (fm-i, 
are the arbitrarily assigned initial constants. 

Substituting in the diHerential equation this value of w, and equating 
coefficients of — 2", wo have 

The expression of the coefficients a„, Om+i, ■■■ in terms of «(,, «!, ,.., "m-i 
depends (by the solution of the foregoing difference-equation) upon the 
algebraical equation 



"When the roots of ^(e)=0 are different from one another, let them be 
denoted by a„ a^, —, a^; and in connection with the m arbitrary constants 
flg, tfi, ..., am_i, determine m new constants ^j, A^, ..., A^, by the relations 



The determination is unique ; for on solving these m. relations as m, linear 
equations in A^, ..., A^, the determinant of the right-hand sides is 



which is equal to the product of the differences of the roots and is therefore 
not zero. Hence, as the constants a^, c,, .... a„_i ore arbitrary, the m new 



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5.] CONSTANT COEFFICIENTS 15 

constants A^, ..., A^, when iised to replace the former set, can be regarded 
aa m independent arbitrary constants. With these constants thus determined, 

= £1 2 on'" + ''-'^K + <'z 2 a^"'*''-^jl^+ ... +c^ S o^'^A^, 
for all values o! n. When n = 0, we have 

2 Ofi"'^^ = CiO„_| + Caa,„,5+ ... +c,„ag = a„^; 
when n= 1, wo have 

and so on, the general result being that 

for all values of n. Hence 

= 2 (J.a,' + ^s''/+ ■■■ +^^<%*)|^ 

the customary form of the solution, A^, ..., A^ being m independent arbitrary 
constants. 

Ex. 7. Apply the preceding method to obtain a similar expression in 
finite terms, when the roots of the equation ij>{6)=0 are not all different from 
one anotiser. 

6. A different method of discussing the linear equation with 
constant coefficients has been given by Hermite. 
Taking the equation, as before, in the form 

we associate with it the expression 

(^) = r" - (Ci?™-^ + c.r™^^ + . . . + c^). 

Denoting \)y f{K) smy polynomial in l^, let 

integration being taken round any simple contour in the i^-plane. 



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16 hermitb's method for equations [6. 

In the first place, the degree of the polynomial /(5') may be 
taken to be less than m. If initially it is not so, then we have 

on division, g (^) being a polynomial, and /, {^) a polynomial of 
order less than that of 0, that is, less than m. Now 



|<!<<,(0<i?-o, 



round any simple contour in the i^-pla-ne ; in the remaining inte- 
gral, the polynomial is of the form indicated. Accordingly, f(X) 
will be assumed to be of order less than m. 
We have 

taken round the same contour ; 



because /(f) is a polynomial and the integral is taken round a 
simple contour in the J;'-plane. Thus TT is a solution of the 
equation. 

The only restriction upon f{^) is that, effectively, its degree 
must be less than m. It may therefore be taken as the most 
general polynomial of degree m — 1 ; in this form, it will contain 
m disposable coefficients which can be used to satisfy the initial 
conditions. Let these conditions require that, when x = 0, the 
variable w and its first m — \ derivatives acquire values ko, hi, .... 
km~-i respectively ; then we determine /(i^) as follows. Since 







we shall dmw the simple contour in the f-plane so as to enclose 
the origin ; and then the preceding relation shews that, when 



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WITH CONSTANT COEFFICIENTS 



^{0 


is expanded it. 


descending 


powers of 


?, 


the coefficient of 


^-r-1 i 


s kr ; so that, as 

m. 
•no 


it liolds for ■ 

^''j + h + 


r = 0, 1, ... 


, w 


i-l. 


we have 


and therefore 














/(?)=•(■«) If +1 + 


.....|? 


+ . 


...}. 





As /(^) is a polynomial in f, all terms involving negative powers 
of ^ must disappear, when multiplication is effected on the right- 
hand side ; and therefore 

/(O ="S^ k, {^^-' - (C.r"'-'^ + .. . + C^r-,)], 

the coefficient of k^-i being unity. If therefore w and its first s 
derivatives are aii to acquire the value zero when z=0, then the 
degree of the polynomial f{^) is m — s — 2. 

In order to obtain the customary expression for W, let the 
contour be chosen so as to include all the zeros of ^(£f). Let a^ 
be a zero, and let its multiplicity be w,, so that 

,(,(f).(!:-.,)".f(r), 

where the roots of ^i (^) are the other roots of ip (^). Let 

/«)_ 41. +_^:-_+ I ^y. ,/■«) 

*«)"?-".(?-«■)■ (f-".)-*.(0' 

jl'ii, -^'ai, ..., being constants, and /i (0 a polynomial of order 
m — Ki — 1. So far as the first tii terms are concerned, their 
contribution to the value of the expression for W is given by 
taking a contour round a, only. We then have 



2lih 



Jf") 



-(r-l)!' ^ 

on changing the constants; and therefore the part, arising through 
the root Ki of multiplicity n-i, in the expression for the integral is 



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18 



hermite's method for equations 



[6. 



involving a number of constants equal to the multiplicity of the 
root. This forra holds for each root in turn ; and therefore the 
number of constants is the sum of the multiplicities, that is, it is 
equal to m, the degree of <l> (f ). But m is the number of arbi- 
trary constants in /(?), when it is initially chosen: these can 
therefore be replaced by the constants A in the expression 

S (jIj + ^=2 + . . . + A„£"-") 6"^ 



the summation extendir 



denoting the 
occurs when a 
another. 



■ the roots a of (j){^) = Q, and n 



lultiplicity of a. The simplest ease, of course, 
the roots of ^{f) = are different from one 



The method can be applied to the equation 



where F(!) ia any function of s. Consider 



where </)((;■ 1 a tl 
in f with (u l,.n v. 
integratioD esteu If 
<l>(0 = 0. Then 



-/••'%?« 



e as b fore / f) 15 a polvnumial 
i coeftic ent^ f he powers ut f, and 
■0 t u tl at icJe-i ill the louts ot 






^™,r,/C,J)«-Oi 









li 



n auceession, until we have 



C^J{^,Odi=0; 



(fo™ 



/{•,0'K-O- 



-/'■ 



■w*+/^)'""r/"'"«- 



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6.] WITH CONSTANT COEFFICIENTS 19 

Heoce, remembering that /(s, is a polynomial in f and that therefore 

we hiive W as a solution of the given equation if, in addition to the other 
cocditioi:iK, which are that 






forj-=2, 3, ..., m, we have 

Now as the contour embraj;es all the roots of <}> {0, we have* 

for r = % .,., m ; ao that, taking 

where d (s) ia a function of z at ouv disposa], we satisfy the ni — 1 formal 
conditions unconnected with F(z) ; and then 6 (?) must be such that 

But as 

«»-3i-,-f»- 
/(•,C)-S'»+5i;./'«""^-f(»)'i«, 



and therefore 
Hence 



where j'(f)is, so far as concerns this mode of determining /(e, f), any function 
of f, and integration with r^ard to u ia along any path that enda in s. When 
F (e) ia zero, / (s, f ) reducea to ff (f) ; and then the solution of the differential 
equation shews that ji(f)is a polynomial in £, of degree not higher than to — 1, 
Aecordii^ly, as ff(^) is independent of i, we take it to be a polynomial of 
degree n^— 1 in f, with arbitrary conatanta for the coefficients ; and then the 
integral of the equation has the form 



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20 CONTINUATION OF THE [6. 

where the f t t t n est nda nd any mple contour including all the 
roots of fji (i)=0 Tj d tl teg at n t nds from any arbitrary initial 

point along J p tl (the pi th b ttcr) to 2. 

Tho ai 1 nt gral th [ n f Wis clearly the complementary 

function, and th d ul 1 mte^ri,! h p t ilar integral, in the primitive of 
the differential equation. The expression can be developed into the customary 
form, in the same way as in the simpler case when ii* (2) vanishes. 

Hermite's investigation, based upon Cauchy's treatment by the calculus of 
residues as espounded in the Exercices de Math^mMiqum, is given in a rnei 
in Darboux's BvXl. des Sciences Math., 2"" S^r. t in (1879), pp. 311—325 : 
followed by a brief note {L c, pp. 325 — 328), due to Dai'bous. A mci 
by Collet, Ann. de I'tc Norm. Sup., 3™ Ser. t. IV (1887), pp. 129—144, may 
also be consulted. 

The Process of Continuation applied to the Synectic 
Integral. 

7. The synectic integral P(z— ^) is known at alt points in 
the domain of J", being uniquely determined by the assigned 
initial conditions at if. So long as the variable remains within 
this domain, the integral at z does not depend upon the path of 
passage from ^ to s, so that the path from f to z can be deformed 
at will, provided it remains always within the domain. Let ^' be 
any point in the domain ; then the values of the integral and its 
first m — 1 derivatives at £" are uniquely determined by the initial 
conditions at f, and they can themselves be taken as a new set of 
initial conditions for a new origin ^'. Accordingly, construct the 
domain of f ' ; and, with the values at f taken as a new set of 
initial values, form the synectic integral which they determine. 
As the new initial values are themselves dependent upon the 
initial values at ^, the synectic integral in the domain of ^' may 
be denoted by Pi (2 -5^. 0- 

If the domain of ^' lies entirely within that of ^ (it then will 
touch the boundary of the domain of ^ internally), the series 
Pi (e — ^', must give the same value as P (z — ^): for every 
point z in the domain of if' is then within the domain of f, and it 
is known that the synectic integral is unique within the original 
domain. 

If part of the domain of if' lies without that of ^, then in the 
remainder (which is common to tho two domains) the series Pi 
must give the same value as P. But in that part which is 
outside, the series Pi defines a synectic integral in a region where 



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7.] SYNECTIC INTEGRAL 21 

P does not exist ; it therefore extends our knowledge of the 
integral, and it is a continuation of the synectic integral out of 
the original domain. 

Let Z be any point in the plane; and join Z ko i^ hy any 
curve, drawn so as not to approach infinitesimally near any of 
the singularities of the coefficients in the differential equation. 
Beginning with if, construct the domains of a succession of points 
along this curve, choosing the points so that each lies in the 
domain of a preceding point and each new domain includes some 
portion of the plane not included by any previous domain. Owing 
to the way in which the curve is drawn, this choice is always 
possible and, after the construction of a limited number of 
domains, it will bring Z within a selected region. With each 
domain we associate its own series ; so that there is a succession of 
aeries, each contributing a continuation of its predecessor. We 
can thus obtain at ^ a synectic integral of the equation, which is 
uniquely determined by the initial values at f and by the path 
from 5" to Z. 

Further, taking the values of the integral and its first m — 1 
derivatives at ^ as a set of new initial values, and taking the 
preceding curve reversed as a path from Z to if, we obtain at if the 
original set of assigned initial values. To establish this state- 
ment, it is sufficient to choose the succession of points along the 
curve in the preceding construction, so that the centre of any 
domain lies within the succeeding domain, and to pass back from 
centre t<) centre. Stating the proposition briefly, we may say 
that the reversal of any path restores the initial values. 

By imagining all possible paths drawn from any initial point if 
to all possible points z that are not singular, we can construct the 
whole region of continuity of the integral, as defined by the 
differential equation and by the initial values arbitrarily assigned 
at if : moreover, we shall thus have deduced all possible values of 
the integral at z, as determined by the initial values at if. It is 
clear, from the construction of the domain of any point and after 
the establishment of a synectic integral in that domain, which 
can be continued outside the domain (unless the boundary of the 
domain is a line of singularity, and this has been assumed not to 
be the case), that the region of continuity of the integral is 
bounded by the singularities of the coefficients. As has already 



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22 DEFORMARLE [7. 

been remarked, these singularities are called the singulanties of 
the equation. Thus all the critical points of the integral are fixed 
points ; and if the equation be taken in the form 






l-?™w 



where the functions q^, ..., q^ are holomorphic over the finite part 
of the plane and have no common factor, these critical points are 
included among the roots of qo, with possibly z = 'Xi also as a 
critical point. The value of the integral at an ordinary point near 
a singularity has been obtained as a synectic function valid over 
the domain of the point, which excludes the singularity. In 
later investigations, other expressions for the integral at the 
point will be determined, when the point belongs to a different 
domain that includes the singularity. 

8. Any path from ^ %o z can be deformed in an unlimited 
number of ways : and it is not inconceivable that these deforma- 
tions should lead to an unlimited number of values of the integral 
at z, as determined by a given set of initial values : but the 
number is not completely unlimited, because all paths from ^toz 
lead to the same final value at z with a given set of initial values at 
£", provided they are deformable into one another without crossing 
any of the singularities. To prove this, consider a path from f to 
z, drawn so that no point of it is within an infinitesimal distance 
of a singularity, and draw a second path between the same two 
points obtained by an infinitesimal deformation of the first; no 
point of the second path can therefore be within an infinitesimal 
distance of a singularity. On the first path, take a succession of 
points z-i, z^, ..., so that 3i lies within the domains of % and of z^, 
% within the domains of z^ and ir,, and so on. On the second path, 
take a similar succession of points a/, si, ..., near Si, i^a, ... respec- 
tively, in such a way that s/ lies in the part common to the 
domains of ^ and s,, while z, is in the domain of zl\ Sj' in the 
part common to the domains of z, and z^, while z^ is in the domain 
oi si; and so on. Join z-iZ-l, z^^, ... by short arcs in the form of 
straight lines. 

Now we have seen that, in any domain, the path from the 
centre to a point can be deformed without affecting the value of 
the integral at the point, provided every deformed path lies within 



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8.] PATHS 23 

the domain. Hence in the domain of ^, the path f^i gives at ^, 
the same integral as the path ^z^'^i. This integral furnishes a set 
of initial values for the domain of Sj ; and then the path S]Zs gives 
at % the same integral as the path s-iS^s^z^. Consequently the 
path fsi^2 gives at z^ the same integral as the path fs/^j, followed 
by z-tZjZ^Zi. But the effect of z^Zi followed at once by z^z^' is nul, 
because a reveraed path restores the values at the beginning of 
the path ; and therefore the path i^z^z^ gives at z^ the same integral 
as the path t^z^ziz^. And so on, from portion to portion; the last 
point on the first path is z, which also is the last point on the 
second path; and tlierefore the path tz,z^...z gives at z the same 
integral as the path t^(z^...z. 

Now take any two paths between if and z, such that the closed 
contour formed by them encloses no singularity of the equation. 
Either of them can be changed into the other by a succession of 
infinitesimal deformations : each intermediate path gives at z the 
same integral as its immediate predecessor: and therefore the 
initial path and the final path from ^ to s give the same integral 
at z ; which is the required result. 

If however two paths between ^ and z are such that the closed 
contour formed by them encloses a singularity of the equation, 
then at some stage in the intermediate deformation the curve will 
pass through the singularity, and we cannot infer the continuation 
along the curve or the deformation into a consecutive curve as 
above. It may or may not be the case that the two paths from 
X,i>a z give at z one and the same integral determined by a given 
set of initial values ; but we cannot assert that it is the case. 

Accordingly, we may deform a given path without i 
the integral at the final point, provided no singularity is c 
in the process. Moreover, in order to take account of different 
paths not so deformable into one another, it will be necessary to 
consider the relation of the singularities to the function represent- 
ing the integral : this will be effected in a later investigation. 

When two paths can be deformed into one another, without 
crossing any singularity, they are called reconcileable ; when they 
cannot .so be deformed, they are called irreooncileable. If two 
irreconcileable paths lead at z to different integrals from the same 
initial values at f, the closed circuit made up of the two paths 
leads at ^ to a set of values different from the initial values. 



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24 FORM OF THE [8. 

These new values can be taken as a new set of initial values : 
when the same circuit is described, they are not restored, so that 
either the old initial values or a further set of values will be 
obtained : and so on, for repeated descriptions of the ciicuit. By 
this process, we may obtain any number, perhaps even an unlimited 
number, of sets of values at ^ deduced from a given initial set ; 
and thus there may be any number, perhaps even an unlimited 
number, of values of the integral at any point e. 

Consider any path from f to s ; and without crossing any of 
the singularities, let it be deformed into loops, drawn from ^ to the 
singularities and back, (these loops coming in appropriate success- 
ion), followed by a simple path (say a straight line) from f to s. 
The final value of the integral at z is determined by the values 
at f at the begiuning of the straight line, and these values are 
deducible from the initial values originally assigned. Hence the 
generality of the integral at z is not affected by taking any particular 
path from f to z, provided complete generality he reserved for the 
initial values : and therefore, from this aspect, it will be sufiBcient 
to discuss the complete system of integrals as arising from com- 
pletely arbitrary systems of initial values at an ordinaiy point. 
This investigation relates to properi^ies of the integrals, which will 
be found useful in discussing the effect of a singularity upon a 
given integral ; it will accordingly be underiiaken at once. 

9. It has already been remarked that the synectic integral, 
determined by the arbitrary constants which are assigned as the 
initial values of the function and its derivatives, is linear and 
homogeneous in those constants: so that, if /a,,, jUia, -.., fiim denote 
the arbitrary constants, and w-^ denotes the synectic integral which 
they determine in the domain of au ordinary point i^, we have 

where ii,, Mj, .... m™ are holomorphic functions of 3 — t^, not involv- 
ing any of the arbitrary coefiicients /i. Take other m — 1 sets of 
arbitrary constants fi, such that the determinant 

I -".I , ^i. Mm , =A(Osay, 



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9.] SYNECTIC INTEGRAL 2-5 

is different from zero. Each set of m constants, regarded as a set 
of initial values, determines a synectic integral, in the domain of 
^; as the quantities v^, u^, ..., u^ in the expression for Wi do not 
involve the arbitrary constants determining w^, it is clear that the 
expressions for these other m — 1 integrals are 

Let Msi denote the minor of jist in the non-vanishing determinant 
A(?); then from the expressions for the m integrals w,, ..., w^ ^"^ 
terms of Wi, ..., u^, we have 

^.{^)u=^MuVh + M^t'a}-,-^--+M^ty'm, (t=l,...,m). 
Now any other synectic integral, determined in the domain of ^ 
by assigned initial values $i, B^, ..., 5,„, is given by 



where the constants & are given by 



:«,! 



(r-1, 



».). 



These constants S- cannot all vanish, when the constants ^i, 9^, ..., 
6m are not simultaneous zeros : for the determinant of the minors 
Mri is {A (?)|™~', and therefore is not zero. Accordingly, any 
integral can be expressed as a linear combination of any m 
integrals, provided the determinant of the initial values of those 
m integrals and their first m — 1 derivatives does not vanish. But 
it is not yet clear that the integrals w-^, ..., w,„ are linearly inde- 
pendent of one another; until this property is established, we 
cannot affirm that the expression obtained is the simplest obtain- 
able. 

Consider therefore, more generally, the determinant of the m 
integrals and their first m~l derivatives, not solely at f but for 
any value of s in the domain of ^, say 



A{z)^. 



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26 A 


PECIAL 


DETERMINANT 


[9. 


When 2= f, it becomes the determinant of initial values denoted 


by a (0- We have 




d^^^) 


d'"iu, 


d™'-'^'W-i 




dt - 


d^ • 


dz'^i *"' 






d^Wi 


rf^-^i/l. 






d!f • 


d^" "'■ 






d"*wm 


i— %« 






-T^- 


d»»-- "' 




=P.aW 




on substituting tor ^,..., 


~^-™ their values in terms of the 



derivatives of lower orders as given by the equation. Hence 

Now within the domain of i^, the function jj, is regular, being of 
the form P,{z — ^); hence the integral in the exponent of e is of 
the form R{z — t,), where it is a regular function that vanishes 
when z=^t. Consequently the exponential term on the right-hand 
side does not vanish at any point in the domain of 5'; also A(f) 
is not zero ; so that A (z) has no zero within the domain of f. 
Moreover, each of the quantities w^, ..., w„ is a holomorphic 
function of z — ^ in that domain, so that A{z) is holomorphic 
also; hence A(3) has no zero and no infinity within the domain 
of the ordinary point C 



i (z) may vanish 
a any region of 



Ak a matter of fact, the only points where 
become infinite are the singularities of pj. For 
3 of the functions w,, ..., !(i,„, we have 



4j,)_ jji* 

4(0 
the path from f to s lying within that region, v/hile s is not i 
the domain off. If a be one of the singularities of pi, the expression of pj i 
any part of an annular region round a as centre is of the form 



where the number of terms in 
according as the singularity ii 



f s-a is finite or infinite, 
jntial ; and g (s) is hoio- 



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9.] FUNDAMENTAL SYSTEMS 

morphio in the vicinity of a. Taking the simplest c 
ix^=a^= ... =0; then 



.'=•*-(!-)>- 



shewing that a ia a zero of A (2) if the real part of a, be positive, and that it 
is an infinity of A iz) if the real part of a, be negative. More generally, the 
nature of A (2) in the vicinity of any singularity a depends upon the character 
of Pj in that vicinity : in tte case of the above more general form, a is an 
essential singularity of A (2). 



Fundamental Systems of Integrals. 

10. The linear independence of w,, ..., Wm, and the property 
that A {£) has a finite non-zero value at any point in the plane 
■which is not a singularity of the equation, are involved each in 
the other. 

It is easily seen that, if a homogeneous linear relation between 
Wi, ..., win, of the form 

CiWi + . . , + c^'.y™ = 

were to exist, the quantities c,, ..., c^ being constants, then A {z) 
would vanish lor aii values of s. The inference is at once 
established by forming the m — 1 derived equations 

and eliminating the m constants Ci,..., c,„ between the m equa- 
tions which involve them linearly: the result of the elimination is 

A(.)-0. 
Hence if, for any set of integrals Wj,..., w™, the determinant A{3) 
does not vanish (except possibly at the singularities of the 
equation), no homogeneous linear relation between the integrals 
exists. 

To establish the inference that, if A (a) does vanish for all 
values of z, a homogeneous linear relation between Wi, ..., w^ 
exists, we proceed as follows. 

In the first place, suppose that some minor of a constituent in 
the first column of A {«), e.g. the minor of —j -J^ in A {z), say 



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28 A FUNDAMENTAL SYSTEM [10, 

Ai ie), does not vanish for all ordinary values of 2 ; and take m 

quantities j/i, ..., ym, the ratios of which are defined by the 
relations 



From the hypotheses that A (s) = and that Ai does not vanish, it 
follows that 



' ds""-' 






Because of the assumption that A^ does not vanish, the ratios 

2/™' ym' '"' 2/m 
are determinate finite functions of s. 

Differentiate the first of the relations: then, using the second, 
we have 

j>i + ...+)/Jw« = 0, 

where 3// denotes dy-ffdz, for the n values of r. Differentiating 
the second of the relations, and using the third, we have 
, dw, , dw,„ 

and so on, up to 

obtained by differentiating the last of the postulated relations 
and by using the deduced relation. We thus have m — 1 relations, 
homogeneous and linear in the quantities y-^, .--, y^ \ in form, 
they are precisely the same as the m — 1 relations, which are 
homogeneous and linear in the quantities j/i, ..., i/,„. Hence, as 
A, does not vanish, we have 

*=.&',, (,.= 1,2 m-l). 



;©= 



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10.] AND ITS DETERMINANT 29 

SO that 

— -^constant = ,—''■, (j- = 1, 2, .... m — 1), 

where Xi, ..., \a~i> ^m ^^^ simultaneous values of y,, ..., ym-\> y-m 
for any particular value of s : that is, the quantities X are con- 
stants. This particular value of s is at our disposal; we may 
assume that X™ is different from zero, because the ratios of ^i , . . . , 
i/ni_, to ym are determinate and finite. Now 

hence 

XlW, + . . . 4- X^Wm — 0, 

that is, a linear relation exists among the quantities w, if A (z) is 
zero, and some minor of a constituent in the first column does not 
vanish. 

Next, suppose that the minor of every constituent in the first 
column vanishes : in particular, let Ai {z) = 0, for all ordinary 
values of z. Then A, (z) is a determinant of m — 1 rows and 
columns, constructed from m — 1 quantities Wi, ..., w„^^ in the 
same way as ^{z), a determinant of m rows and columns, is 
constructed from the m quantities w,, ..., Wm- The preceding 
analysis shews that, if some minor of a constituent in the first 
column of Ai (z) does not vanish for all ordinary values of z, then 
a relation 

where k, ,...,«ni-i ^^^ constants, is satisfied: so that a linear 
relation exists among the quantities w, and it happens not to 
involve ic^- 

Lot the process of passing from A {z) to A, (s), fi-om A, {z) to a 
corresponding minor, and so on, be continued; the successive 
steps are effected by removing the successive columns in A{3) 
beginning from the left and by removing a corresponding number 
of rows. At some stage, we must reach some minor which is not 
zero tor all ordinary values of e : so that 



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30 THE NUMBER OF [10. 

vanishes whea s = 0, 1, ..., r, but is different from zero when 
s = r + l. Then the earlier analysis shews that a linear relation 
of the form 

/JlW, + . , . + pm^VJr„^, = 

exists, where pj,..,,p„^, are constants: in effect, a linear homo- 
geneous relation among the quantities Wi, ..., Wm which happens 
not to involve Wm-i+i, ■■-, w™- Hence, if the determinant A(^), 
constructed from ike m integrals w^, ..., w^, vanishes fiyr all 
<yrdinary values of z, there is a komoffeneo'us linear relation between 
these integrals. 

Integrals are sometimes called independent when they are 
linearly independent, that is, connected by no homogeneous linear 
relation ; but the independence is not functional, because all the 
integi-als are functions of the one variable z. A set of m linearly 
independent integrals w is called a fundamental system. ; and each 
integral of the set is called an element or a member of the system. 
The determinant A (a), constructed out of a set of m integrals, is 
called the determinant of the system; so that the preceding results 
may be stated in the form ; — 

If the determinant of a set ofni integrals vanishes for ordinary 
(that is, non-singular) values of the variable, the set cannot constitute 
a fundamental system ; and the determinant of a fundamental 
system does not vanish for any non-singular value of the variable. 

11. We now have the important proposition: — 

Every integral, which is determined by assigned initial values, 
can be expressed as a homogeneou^s linear combination of the 
elements of a fundamental system. 

Let W denote the integral determined by the assigned values 
at if, taken to be an ordinary point of al! the coefficients in the 
differential equation ; and let w, , . . . , Wm he a fundamental system. 
Let constants c,,...,Cm be deduced such that, when 3 = ^, we 
have 

W= 2 CaWa \ 



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11.] LINEARLY INDEPENDENT INTEGRALS 31 

This deduction is uniquely possible; because the determinant of 
the quantities c on the right-hand aides is the determinant of a 
fundamental system, and therefore does not vanish when z=^. 

Thus W ~ i, Ct,WK is an integral of the equation; this integral 

and its first m — 1 derivatives vanish when 3=^; so that it 
vanishes everywhere (Cor, I, § 5), and therefore 

the constants c being properly determined as above. 

Coe. I. Between any m + 1 branches of the general solution, 
there must be a homogeneous linear relation. For if m of them be 
linearly independent, the remaining branch can be regarded as 
another integral : by the proposition, it is expressible linearly in 
terms of the other m. 

Cor. II. Any system of integrals u^, ..., «,„ is fundamental if 
no relation exists of the form 



where A^, ..., A^ are constants. For taking a fundamental system 
W], ..., Wm, we can express each of the solutions u in the form 

Mr = «lr«'l+ ■■■ + IhnrVm. (r=l, 2, .,., m), 

where the coefficients a are constants. If G denote the determ- 
inant of these m^ coefficients, C must be different from zero: for 
otherwise, on solving the m equations to express Wi in terms of 
Ml, ,.., Mfli, we should have a relation of the form 

A^Uj+ ... + .4„M„ = Cwi = ; 
and no such relation can exist. If, then, A^ (e) denote the deter- 
minant of the set of integrals u, and if A„ (s) denote that of the 
fundamental system w^, ..., Wm, we have 
A.W-CA,W, 

by the properties of determinants. Now C does not vanish, nor 
does A,j (s) at any ordinary point in the plane ; hence A„ (2) does 
not vanish at any ordinary point in the plane, and therefore 
Ui, ...,u,a ^^e a fundamental system of integrals. 



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32 A SPECIAL [11. 

The result may be stated also as follows : If m integrals u be 
given hy equations 

u =a,iL,+ +« ' „ <<=1 m) 

where the deteimmant of the coe_ffnienfs a i/, nd ^eio and t/e 
integrals w are a fjnda mental system then the bystem of mteQiaU 
u is also fundamental 

12. One paiticuHi fundamental sjstem for the difteiential 
equation can be jbti.ined as follows Let w be a sppciil mtegial 
of the equation that is an integral deteimioed 'h^^ an> special set 
of initial conditions, and substitute 

w = wjvde 
in the e(juation : then v is determined by the equation 



Similarly, let v^ be a special integral of this new equation, with 
the appropriate conditions ; then substituting 

V = vjudz, 

we find that the equation, which determines u, is of the form 

where 

m-ldv, 

'■^ = ^^ ^rf7- 

And so on. 

It is manifest that the quantities 

w,, wjv^dz, w,f(vjihds)ds, ... 

are integrals of the original equation. Moreover, they constitute 
a fundamental system ; for, otherwise, they would be linearly 
connected hy a relation of the form 

CiW, + dwjvjds + C3W,j{vJu,dz) dz+ ... = 0, 
that is, , 

Cj + c.jjv,dz + cj(vju,dz) dz+ ... = 0. 



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12.] FUNDAMENTAL SYSTEM 33 

When this is differentiated, it gives 

Cj-Ui + Civjujds + ... = 0, 
that is, 

c.i + c,Juidz + ... = 0. 

Effecting m — 1 repetitions of tliis operation of differentiating and 
removing a non-zero factor, we find 

as the result at the last stage. Using this in connection with the 
equation at the last stage but one, we have 

c™-, = 0. 
And so on, from the equations at the various stages, we find that 
all the coefficients c vanish. The homogeneous linear relation 
therefore does not exist : the system of integrals, obtained in the 
preceding manner, is a fundamental system. 

As an immediate corollary from the analysis, we infer that 
^i. Vifu,dz,... 
constitute a fundamental system for the equation in v ; and so for 
each of the equations in succession. 

The determinant of this particular fundamental system is 
simple in expression. Denoting it by A, and denoting by Ai the 
determinant of the fundamental system of the equation in v, 
we have, as in § y, 



^Pi' 



IdA 
i^'dz 
1 dAi _ _ m dwi 



A dz Ai dz ' 



where Xi is a constant. Similarly, if A3 denote the determinant 
of the fundamental system of the equation in u, we have 



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34 FORM OF THE DETERMINANT [12. 

and so on. The last determinant of all is the actual integral of 
the last of the equations ; hence 

i-Cwi™?)j"'-'u,™-^.., 

where (? is a constant. Moreover, A is the determinant of a par- 
ticular system, so that C is a determinate constant. It is not 
difficult to prove that 

and therefore 
conaequtntly, 

Sx. Verify the last result, as to the form of A, in the case of 
(i) Legendre's equation : 

(ii) the equation of tte hjpei^eometric series : 
(iii) Bessel's equation. 



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CHAPTER II. 



General Form and Properties of Integrals near a 
Singularity. 

13. We have seen that, within the domain of an ordinary 
point, a synectio integral of a linear differential equation is 
uniquely determined by a set of assigned initial values ; and that 
the said integral can be continued beyond that domain, remaining 
unique for all paths between the initial and the final values of the 
variahle which are reconciteable with one another. When the 
variable is permitted to pass out of its initial domain though 
returning to it for a iinal value, or when two paths between the 
initial and the final values are not reconcile able, the various 
propositions that have been established are not necessarily valid 
under the modified hypothesis : it is therefore desirable to con- 
sider the influence of irreconcileable paths upon an integral, 
still more upon a set of fundamental integrals. Remembering 
that any path is deformable without affecting the integral if, in 
the deformation, it does not pass over a singularity, we shall 
manifestly obtain the effect of a singularity, that renders two 
paths irreconcileable, by making the variable describe a simple 
circuit, which passes from the point z round the singularity 
and returns to that point z, and which encloses no other 
singularity. 

Let a be the singularity round which the simple closed circuit 
is completely described by the variable. Let m;,, ..., in^ denote a 
fundamental system at z ; and suppose that the effect of the 

3—2 



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36 EFFECT OF A SINGULARITY [13. 

circuit is to change the m integrals into w/, ..., Wm respectively. 
That the set of m new integrals thus obtained is a fuudamental 
system can be seen as follows. If it were not a fundamental 
system, some relation of the form 

2 k,w; = 

would exist, with constant coefficients k, for all values of z in the 
immediate vicinity. In that case, the quantity S krW,' (which is 
an integral) is zero everywhere, together with all its derivatives, 
as it is continued with the variable moving in the ordinary part 
of the plane. Accordingly, let the integral be continued from z 
along the closed circuit reversed until it returns to z where, by 
what has been stated, it is zero. The effect of the reversal is 
(§ 7) to change w/ into w, : and so the integral after the reversed 
circuit has been described is % krWr, so that wc should have 

2 Kw, = 0, 

contrary to the fact that v\, .,,, w^ constitute a fundamental 
system. The initial hypothesis from which this result is deduced 
is therefore untenable : there is no homogeneous linear relation 
among the quantities w/, ..., Wm, which therefore form a funda- 
mental system. 

Since the system Wi, ..., Wm is fundamental, each of the inte- 
grals w/, ..., Wm is expressible linearly in terms of the elements 
of that system ; so that we have equations of the form 

w/ = a:siiy,+ ...-l-«^w™, (s=l, ...,m), 

where the coefficients a. are constants. As the system Wg is 
fundamental, the determinant of these coefficients is different 
from zero : this being necessary in order to ensure the property 
that W], ..., Wm are expressible linearly in terms of w/, ..., Wm, a 
fundamental system. 

Take any arbitrary linear combination of the system, say 

where the coefficients p are disposable constants ; and denote this 
integral by u. When the variable desci'ibes the complete closed 



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13.1 



UPON A FUNDAMENTAL SYSTEM 



3Y 



circuit round the singularity, let w' denote the modified value of 

u, so that 

U = pjWi + . . . + pmlOm' 



= Pi 2 a,rW, 



. + pml a^rWr- 



It is conceivable that the coefficients p could be chosen so that 
the integral reproduces itself except as to a possible constant 
factor; a relation 



would then be satisfied, 6 being a constant quantity. This rela- 
tion, in terms of w,, ..., w™, is 

Pi 2 a^rWr^ ... + pm 2 a^rWr 
= 9 (piW, + ... + p™Wm), 

which, as it involves only the members of a fundamental system 
linearly, must be an identity: the coefficients of w,, ..., w™ must 
therefore be equal on the two sides. Hence we have 



Pittis + p2 (0=2 - ^) 4 



■ + PmCtmi 

■ ■ + pm«mi 



If, therefore, $ be determined as a root of the equation 



the preceding relations then lead to values for the ratios of the 
constants p for each such root. It is to be noted that, in this 
equation, the term, which is independent of $, does not vanish, 
for it is the determinant of the coefficients a ; hence the equation 
has no zero root. 

As the equation definitely possesses roots 9, it follows that 
integrals exist which, after a description of the simple contour 
round a, reproduce themselves save as to a constant factor. If it 
should happen that the constant factor is unity, then the effect of 



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38 THE FUNDAMENTAL EQUATION [13. 

description of the contour upon the integral is merely to leave it 
unchaiiged : in other words, such an integral is uniform in the 
vicinity of the singularity. 



Propeuties of the Fundamental Equation. 

14. The special significance of the equation, in relation to 
the singularity a, lies in the proposition that the coefficients of ike 
various powers of 6 in A =0 are independent of the fundamental 
system- initially chosen for discussion. To prove* the statement, 
it will be sufficient to shew that the same equation is obtained 
when another fundamental system is initially chosen. For this 
purpose, let y,, ,,.,ym denote some other fundamental system; 
and suppose that, by the simple closed contour round a described 
by the variable, the members of the system become j^/, ,.,, ym 
respectively. Then, as both these systems are fundamental, there 
are relations of the form 

y/ = 0s,yi + -- + 0^y^, (s = i, ....m), 

where the determinant of the coefficients /3 is not zero. The 
equation B = 0, corresponding to j1 = for the determination of 
the factor 8, is formed from the coefKcients ^ in the same way 
as A from the coefficients a, so that the expression for B is 

B = 



Because each of the sets w„ ,.., w^; i/,, ,.,, i/^; is a funda- 
mental system, the members are connected by relations of the 
form 

y., = y„w, + y^w^+... -1-7™^^, (r^l, ..., m), 

where the determinant of the coefficients, which may be denoted 
by r, is different from zero. The quantity 

is zero everywhere in the vicinity of z ; and it is an integral, 
which accordingly is zero everywhere in its continuations over the 

* The proof adopted is due to Hamburger. Crelk, t.'Lxxvi (1873), pp. 113—125. 



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14.] IS INVARIANTIVE 39 

ordinary part of the plane. When it is continued along the 
simple contour round a, the variable returning to z, the integral 
is zero there ; that is, 

Hence 



and therefore 



S 2 /SrsJeiWi =22 7rs«siW[. 



This relation involves only the members of a fundamental system 
hnearly ; hence it must be an identity. We therefore have 

2 ^r67B(= 2 7rsas* 



say, the relation among the constants holding for all values of r 
and t. Now forming the product of the determinants P and A, 
we have 



7n. Tis' 7i3. 

721. 722. 7S3' 

7si, 7aa, 7as. 



5ii-7u^. Sk-7iA ... 
821 — 7h^, S22 — 7,2^, ■ ■ ■ 

say ; and similarly, forming the product of B and V, we have 
' — 6, ^,2 , /3i3 , .■■ 7ii, 7si, 731. 
, ji^-0, /3a , ... 7i3. 7^. 732. 
, ^^ ,0x1 — 0. ■■■ 7i3> 723. 73s. 



-7n^. S,.,-7,A ... 

-72A ^^-y^0, ... 



= -0, 



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40 INVAEIANTIVE PROPERTIES [li. 

identically. Also F does not vanish ; hence 

for all values of B. 

Accordingly, the equation A = is invariantive for all funda- 
mentaJ systems in regard to the effect of the singularity a upon 
the memhers of the system: it is called* the fundamental equation 
belonging to the singularity a. We note that its degree is equal to 
the order of the differentia! equation. 

While the equation is thus invariantive for all fundamental 
systems, the actual invarianee of one of its coefficients is put in 
evidence, either when the differential equation of § 2 is initially 

devoid of the term involving -j-;^ , or after the equation has 

been transformed by the relation 

SO as to be devoid of the term involving , ^_^ . In A =0,the 

term, which is independent of 6 is equal to unity, a property first 
noted by Poincar^f. For when ^i is zero, the determinant A of 
the fundamental system is a constant, for (§ 9) its derivative 
vanishes ; it therefore is unchanged when the variable describes 
a simple closed circuit round the singularity. The effect of such 
a circuit upon A is to multiply it by the term in A which is 
independent of 8 : accordingly, that term is unity. 

The linear equation can always be modified so that the term 
involving the derivative of the dependent variable next to the 
highest is absent; and the necessary linear modification of the 
dependent variable leaves the independent variable unaltered. 
This change does not influence the law giving the effect, upon 
the integrals, of a description of a loop round the singularity; 
and the fundamental equation is independent of the choice of 
the fundamental system. Accordingly, the coefficients of the 
various powers of 9 (except the highest, which has a coefficient 
(—1)™, and the lowest, which has a coefficient unity) are fre- 
quently called the invariants of the singularity : they are m — 1 
in number. 

• Sometimes also tha charaeteristic equation. 
t Acta Math., t. iv (1884), p. 202. 



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15.] OF THE FUNDAMENTAL EQUATION 41 

15. There is a further important invariantive property of the 
determinants A(d), B{ff), viz.: If all minors of order n (and 
therefore all minors of lower order) in A (8) vanish for a particular 
value of 6, but not all those of order n + 1, then all minors of order 
n in, B {&) also vanish for thai, value of 8, hut not all those of order 
n + \. 

A minor of order n is obtained by sappresaing n rows and n 
columns ; accordingly, the number of them is 



say. Let them be denoted by ftfj, hij, c^, dij when formed from 
A {0), B (0), r, J) respectively, where * and j have the values 
1, ..., /i, these numbers corresponding to the various suppressions 
of the rows and the columns. Then, regarding D as the product 
of A and V, we have* 

dij = Ci,aj, + Gjaaj-i + . . , + Ci^aj> ; 
and regarding D as the product of B and T, we have 

dij = biiCj, + bi^Cji+ ... +i'i„Cj„. 

All the quantities (ifj are supposed to vanish for a particular value 

of d ; hence for that value all the quantities dij vanish. Assigning 

to j all the values 1, ..., ^ in turn, we therefore have 

= Cii&ii +Cis&i3 + ... +c,^bi^ \ 

=- cAi + Cs^bi2 + ... +c.^bi^ I , 



= C^ibij + C^s6;s + . . . + c^^bi^ I 

The determinant of the coefficients of bi,, hi,, ... , V '^ equal tof 

T\ 
where 

x = _C^-i)'_. 

{m-n-\)\ nV 
that is, the determinant does not vanish. Accordingly, we must 
have 

ht^^O, b^ = (},...,hi^ = 0; 

as this holds for all values of i, it follows that all the minors ot 
B{9) of order n vanish for the particular value of 0. 

* Scott's Determinants, p. 53. + ih., p. lil. 



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42 ELEMENl'ARY [15. 

The minors of B (S), which are of order li + 1, cannot all vanish 
for the value of 0; for then, by applying the result just obtained, 
all those of A (0), which are of order n+1, would vanish, contrary 
to hypothesis. 

16. A more general inference can bo made. Leaving arbi- 
trary and not restricting it to be a root of the fundamental 
equation, the two expressions for dij give 

holding for all values of i andj. Taking this equation for any one 
value of _y and for all the /t values of i, we have ^ equations in all, 
expressing a,-,, a^, ..., iij> linearly in terms of bpq. The determi- 
nant of coefficients on the left-hand side is T*, as before, and does 
not vanish ; so that each of the quantities Oj^ is expressible linearly 
in terms of the quantities ftp,, the coefficients involving only the 
constituents of V. Similarly, taking the equation ibr any one 
value of i and for all the ^ values of j, we find that each of the 
quantities bp^ is expressible linearly in terms of the quantities aj>, 
the coefficients involving only the constituents of F. If therefore 
all the quantities a^r have a common factor — 0i, and if that factor 
be of multiplicity o-, then all the quantities bpq also have that 
factor common and of the same multiplicity a- ; and conversely. 

These results associate themselves at once with Weierstrass's 
theory of elementary divisors*. If {0 — Oi)" is the highest power 
of 8 — 0-, in A (6), if (6 — O^y^ is the highest power of that quantity 
common to all its minors of the first order, if (0 — 01)"' is the 
highest power common to all its minors of the second order, and 
so on, then (as will be proved immediately) 

a >a;><7^>...; 
and 

(«-»,)—', (9- e,)-.--., ... 

are called elementary divisors of the determinant A (0). It follows 
from the preceding investigation that the elementary divisors of 
the fundamental equation, are invariantive, as well as the equation 

* Berl. Monatsber., (1868), pp. 310—388; Oes. Werke, t. ir, pp. 19—44. See 
aUo & memoir by Sauvage, Ann. lie l'£c. Norm., 3" S&-., t; vm (1891), pp. 26a— SiO ; 
and a treatise by Math, EtemenlaTtlieiUr, (Leipzig, 1899). 



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16.] DIVISORS 4-3 

itself; for they are independent of the particular choice of a 
fundamental system. 

If the earliest set of minors of the same order that do not all 
vanish when = Si is of order p, so that they are of degree m.~ p 
in the coefficients in A, then the elementary divisors are 

being p in number : and then p is one of the invariantive numbers 
associated with the particular singularity of the equation. 

As two of the properties of the invariantive equation, associated with the 
elementary divisorB, are required, they will he proved here : for full discusaion 
of other properties, reference may bo made to the authorities quoted. 

It is easy to obtain the result 

cr>o-i>irj> ..,, 
just stated above. For 

9^ Z A 

M = -.!/- 

wlieie -J r 1^ the muioi ot a^, 6 In A„ there is a factor (0-0,)"', for each 
of tlie qmiititie-i ■!„ is i fir«t minor ; therefore that factor occurs in their 
sum and, owm^ to the uimbination of terms, it may have an even higher 
indes than o-, On the left, the factor in ^ - ^j has the index o- - 1 i hence 

that is, 

Similarly for the other inequalities. 

Again, we know • that any minor of degree p which, can be formed out of 
the first minors of A {&) is equal to the product of ^p^' {6) by the comple- 
mentary of the corresponding minor of A {6). Hence, taking p = 2, we have 
relations of the form 

A^B2-A^B^ = AC, 

ora of the first order, and C is a minor of the 
r of the second order which is divisible by no 
higher power of 6— 6, than (fl— fl,)"'; the left-hand side is certainly divisible 
by {6 - S^f"', and it may be divisible by a higher power if the terms combine ; 
hence 

that is, 

Similarly, we have the other inequalities of the set 

cr-o-^>T^-ir^5:a2-irs>...><ri,_i, 
so that the indices of the elementary divisors, as arranged above, form a 
series of decreasing numbers. 



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BRANCHES OF AN ALGEBRAIC FUNCTION" 



[17. 



Association of Differentiai- Equations with Algebraic 
Functions. 

17. Before considering the roots of the fundamental equation, 
it is worth while establishing a converse* of the propositions in 
§ 13, as follows: 

Let j/i, ,.., ym he m linearly independent functions of s, which 
are uniform over any simply-connected area not including any 
critical point of the functions : let the critical points be isolated and 
let each of them be such that, when a simple contour enclosing it is 
described, the values of the Junctions at the completion of the contour 
are given by relations of the form 

yr'=a«,?i + --. + «™2/m, ('• = 1, ...,m), 

where the determinant of the coefficients a. is not zero, and the 
constants may cJiange from one critical point to another : then, the 
m functions are a fundamental system of integrals for a linear 
differential equation of order m unth uniform coefficients. 

It is clear that, if the functions are integraSs of such an equation, 
they form a fundamental system because they are linearly indepen- 
dent. On account of this linear independence^ the determinant 



dz'"^' ' 



d^"^" 



d^^' ' dz"'-' ' ■■■■ 
does not vanish for all values of z. Let As denote the determinant 
which is derived fi-om A by changing the stii column into -j—^ , 
'"' rir^ ' ^^^ consider the quantity 

For any contour that encloses no critical point, A and A, are 
uniform, so that ps is uniform for such a contour. For a simple 
contour, which encloses the critical point a and no other, the 

' It is given by Tannery, Ann. de VKc. Norm.. Ser. 2"", t. iv (1875), p. 130. 



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17.] AS A FUNDAMENTAL SYSTEM 45 

determinant A after a single description acquires a constant factor 
R, where R is the (non-zero) determinant of the coefficients in the 
set of relations 

^/r' = «rii/i+--- + a™3/m, (r = 1, ...,m). 

The determinant Aj acquires the same factor R, in the same 
circumstances ; and therefore ps is unchanged in value by a 
description of the contour, that ia, it is uniform for such a 
contour. As this holds for each contour, it follows that pg is 
uniform over the plane. 

The m quantities y,, ..., y^ evidently are special integrals of 
the equation 

which is linear and the coefficients in which have been proved 
uniform functions of s. 

Corollary. If all the critical points of the functions are of 
an algebraic character, that is, of the same nature as the critical 
points of a function defined by an algebraic equation, and are 
limited in number, then the uniform coefficients p m the differential 
equation are rational functions of z. For as p^ is uniform, the 
critical point a is either an infinity, or an ordinary value (including 
zero). If it is an infinity, it can be only of finite multiplicity; 
for the critical point is one, where A and A, can vanish only to 
finite order because of the hypothesis as to the nature of the 
critical point: that is, the point is then a pole of finite order. 
Likewise, if it is a zero, the multiplicity of the zero is finite. 
This holds at each of the critical points of the functions y\,..-,ym\ 
and the number of such points is finite. Moreover, every point 
that is ordinary for each of the functions is ordinary for A and Ag 
and, in particular, A cannot vanish there : so that no such point 
can be a pole of any of the coefficients p. It therefore follows* 
that each of these coefficients is a rational meromorphic function 
oiz. 

The converse of the corollary is not necessarily (nor even 
generally) true : it raises the question as to the tests sufficient 
and necessary to secure that the integrals of a linear equation with 
rational coefScients should be algebraic functions of the variable. 
This discussion must be deferred. 

* T. F., S 43. 



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ALGEBRAIC FUNCTIONS 



[ir. 



46 

Ex. 1. The most conspicuous instance arises when the dependent vari- 
able w is an algebraic function of z, defined bj an algebraic ec[iiation 

of degi'ee m io w. Ba«h branch of the function so defined is uniform in the 
vicinity of an ordinary point ; in the vicinity of a branch-point, the branches 
divide themselves into groups ; and any linear combination of them is subject 
to the foregoing laws of change (which take a particularly simple form in this 
ca.se) when z describes a circuit round a branch-point. 

To obtain the homogeneous linear equation of order m which is satisfied 
by every root of j''=0, we can proceed as follows. Let ^(s) = be the 
eliminant of /=0 and =~=0; so that* all the branch-points of the alge- 
braic function are included among the roots of 0=0, though not every root is 
a branch-point. By a result f in the theory of elimination, we know that the 
resultant of two quantics u and v of degi'ee ni. and n respectively in a variable 
to be eliminated is of the form 

where Mj and % are of degrees m- 1, «— I respectively in that variable ; and 
therefore 



where i/ is of degree m— 2 io w and V is of degree ni- I in n; But / is 
? equal to zero for all the values of w considered ; hence 



.ally derivable from Sjl 




of the ehminant, ( 



To the last column, add the first column multiplied by .r™"'^"^, the second 
multiplied by iii'"+"-2, and so on : a change which does not affect the value of E. 
The couBtituants in the new last column ace 

x'^hi, x'^ht, .... xu, K, x^^-^i), i™-2i), .,., XV, v; 
eipanding E by taking every term in this last column with its minor, eoUeo ting all 
the terms involving ti into one set and those involving v into naother, we have 



where v, 









s of degree m -1 ii 



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17.] 



AND DIFFERENTIAL EQUATIONS 











By m.» 


i of/=0 which ia of degree m 


in w, we can 


, reduce v'f- 


contains i 


10 power of w h^her than the (i 


m-I)th, say 





where P, is a polynomial in w of degree not higher than m-1. (If the 
highest term in / has unity for its coefficient, then P, ia a polynomial in s 
also.) Aj;ain, 

d^ Pi dp, 1 dP, F, a^ 

d^a 02(3) ^ -+-0(2) g^ 02(5) a^ 



on reduoing to a common denominator ; by u 
Pj can he made of degree not higher than nj- 
uniform functions of z. And so on, up to 



where I'^ is a polynomial in i 
being uniform functions of e. 



ns of /=0, the polynomial 
1 M, and its coefficients aro 



7 of degree not higher than m - 1, the coofflcionta 
We thus have 



Among these m equations t 
m- 1 quantities u^", a)*, ..., iP" 
the form 



can, by a linear combination, eliminate the 
■^ irom the left-hand aides ; and the result has 



&•«=' 



™^2m 9" 



where Q„, <2i, ..., §m are uniform functions of s, Thia is satisfied for every 
root JO of the algebraic equation : and it is of order m. 

Corotlary. There ia one special case, when the differential equation is of 
order m - 1, viK., when the algebraic equation ia 

/=w"'-l-a2)«™-=+ ... +0^=0, 
BO that the term in w™~' is absent. We then have 



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48 EXAMPLES OF [17. 

SO that one of tho in, branches w can he espreaaed ]inearly in terms of the 
others ; Tannery's r^ult ahewa that the differential equation is theii, of order 
•not higher than m — 1. In that case, it would be sufficient to take only the 
m — 1 equations 

^-1"*'. (- •»-')• 

For instance, consider the algebraic equation 
-u^ + to^M, 
where u is any function of 2 ; it is to be expected that the liaear differential 
equation satisfied by each of the three branches of the function defined by 
this cubic equation will be of the second order, say 

where A and B are functions of :. We have 

'-'+')^+*'(J)"-i»- 

so, substituting in 

(».+i)5' + JK+i)f+^(«'+«)-o, 

and using M'^ + 3;(i=ii, wo have 

Multiplying the right-hand side by {«j* + 1)^, and the left-hand side by its 
equivalent l + wu — vfi, we have 

on reduction by the original algebraic equation. This will hold for each of 
the three roots of that equation, if 

g«'^=«(iJ«.'+K)+^(-«^-8)l 
O^lA'd + 1%" + 'ABu y 

These conditions give the values of A and B ; and the equation for w is easily 
found to be 



dho ( uu- u"\d-w_ u'^ 
"^2"'"UH4 u-)dz *ii«+4'^ 



where u' and u" are the first and the second derivatives of u. The equation 
oi the seco 1 de as d ted 

A 1 When the Igeb a equat on of degree m is of quite 

g ner 1 fo m the 1 near 1 fierent al equat n sat sfied by t roots is of order 
m B t when the ^gebra c eq at on has orj sj al form-i, though still 
rre I ble the d fferent al eq at on in y be of order 1 s th m ; for the 



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17.] DIFFERENTIAL RESOLVENTS 49 

elimination of various powers of w may not require derivatives up to that of 
order m. The most conspicuously simple ease is that in which the alge- 
braic equation is 

where K is a rational function of z ; the differentia! equation is 

only of the first order. 

Other cases occur hereafter, in Chapter v, where quantities connected with 
the roots of algebraic equations of degree higher than two satisfy linear 
differential equations of the second, order. 

^ote 2. The differential equations considered have, in each case, been 
homogeceoua. If we admit non-homogeneous linear differential equations, 
viz. those which bave a term independent of w and its derivatives, then in 
the general case, where /(w, i) has a term in (t"""*, the differential eqviOtion is 
of order m—1 ordy. This can be seen at once from the elimination of 
w^, -ui*, ..., a;™-! between 

^ ds ^ 



leading to a (non -homogeneous) linear equation of order m- 1. This result 
appeal's to have been iirst stated by Cockle*; it is the initial result in the 
formal theory of differential resolventst. 

&«. 2. Shew that, when the algebraic equation is 

the two linear differential equations, homogeneous and non-homogoneous 
respectively, are 

^ _ 3-t-23*i rf«j 3+22= 

d^ Z-^r^ dz 1'+^'^ ' 

dm H-2s^ # 

di s+s^ ^'"' 1+^' 
E.V. 3. Obtain the differential equations satisfied by efich root of 

(i) ffj3_3^2^^ = 0; 

(ii) i!^-Zzw^si=0. 

Ex. 4. Shew that any root of the equation 

r-ny = {n-l)x: 

* FUl. Mag., t. sxi (1361), pp. 37fl— 383. 

t Foe cefarences, see a paper by Harley, Manch. Lit. and Phil. Memoirs, t. v 



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50 SIMPLE ROOTS OF [17. 

(n being greater than 2) satisfies tlie eqiiatiuii 

wheiw a = l — . What ia the form for « = 2? (Heymann.) 

Ek. 5. Shew that any root of the equation 

(« lieing greater than 2) satisfies the equation 

where the constants a^ arise as the coeffioients in the algebraic equation 











_s(-iri 


jA'- 


when the roots a 


ro 
















x=(i--,)^^'L^ 


fori = l, . 


.., B-. 


L, and 


«..- 


-1=1- 




Ex.e. 
then 


Prove that, 


if 


r--V+5y- 


-ix 



i^y 2,-g-l 



rT=0; 



and explain the decrease in the order of the differential equation. 

(Math. Trip., Part ii, 1900.) 

^ii^■1)amental system of integrals associated with a 
Fundamental Equation. 

18. We now proceed to the consideration of the fundamental 
equation A = Q appertaining to the singularity a. 

The simplest ease is that in which the m roots of that equation 
are distinct from one another, say 6^, 9^, ..., 6^- Not all the 
minors of the first order vanish for any one of the roots : if they 
did vanish, the root would be multiple for the original equation. 
Hence each root 6r determines ratios of coefficients c^, c,.j, ..., Crm 
uniquely, such that an integral of the equation exists, having the 
value 

and possessing the property that 



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18.] THE FU^DAMENTAL EQUATION 51 

where m/ is the value of Ur after z has described a complete simple 
contour round a. We thus obtain a set of m integrals. 

These m integrals constitute a fundamental system : otherwise 
a permanent relation of the form 



KlMl + Kstf, + . . . + K 

would exist. This quantity Sk^w^ is 



= 



integral ; as it is zero 
and all its derivatives are zero at and near z, it is zero everywhere 
when continiied over the regulaj- part of the plane. Accordingly, 
let z describe a simple closed contour round a: when it has 
returned to its initial position, the zero-integral is Sk^m/, that is, 

kA-Hi + icAui +... + K^O,^iim = 0. 
Similarly, after a second description of the simple closed contour, 
we have 

KA'Ur + 'cA^ti, + ... + >c„e„>,^ = 0. 

made in this way : we 



Let m — 1 descriptions of the contour 1 



(i + K 



for r = 0, 1, ... 
zero, we have 



i all the coefficients / 



that is, the product of the differences of the roots is zero. This is 
impossible when the roots are distinct from one another; hence 
the coefficients «,, ..., k^ vanish, and there is no homogeneous 
linear relation among the integrals Wi, ..., «,„, which accordingly 
constitute a fundamental system. 

The general functional character of these integrals is easily 
found. Let 



so that T^ is a new constant, which is determinate save as to any 
additive integer; as the roots 8i, ..., dm are unequal, no two of 
the m constants r^, ..., r^ can differ by an integer. Now the 
quantity 



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52 EFFECT OF A [18. 

acquires a factor e "'''', that is, 6^, when z describes the simple 
complote circuit round a. Hence the quantity 

returns to its initial value after the variable has described the 
simple complete circuit round a; and therefore it is a uniform 
function of s in the immediate vicinity of a, say tf^, so that 

As this holds for each of the integers i^, it follows that we have a 
system nf fundamental integrals in the form 

where tj>,, 02, ..., (f>m o-i'^ uniform functions of z in, the vicinity of 
a, the quantities r^ are given by the relations 

j-„=^log^„, 

and the roots 0,, ..., dm of the fandamental equation are supposed 
distinct from one another, no one of them being zero. 

As regards this result, it must be noted that the functions ^ 
are merely uniform in the vicinity of a : they are not necessarily 
holomorphic there. Each such function can be expressed in the 
form of a series of positive and negative pov^ere of z — a, converg- 
ing in an annular space bounded by two circles having a for a 
common centre and enclosing no other singularity of the equation. 
There may he no negative powers of ^ — a, in which case the 
function >}> is holomorphic at a ; or there may be a limited number 
of negative powers, in which case a is a pole of <^ ; or there may 
be an unlimited number of negative powers, in which case a is an 
essential singularity. Moreover, r^ is only determinate save as to 
additive integers : it will, where possible (that is, when a is not an 
essential singularity), be rendered determinate hereafter ; so that, 
in the meanwhile, the result obtained is chiefly important as 
indicating the precise kind of multiform charactei' possessed by 
the integrals near a singularity. 

19. Now consider the case in which the fundamental equation 
A = appertaining to the singularity a has repeated roots, say Xi 
roots equal to 0,, X^ roots equal to 8^, and so ou^ where ^,, 6^, ... 
are unequal quantities, and Xj + Xj + . . . = m. It will appear that 



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19.] MULTIPLK ROOT 53 

a gi-oup of linearly independent integrals is associated with each 
such root, the number in the group being equal to the multiplicity 
of the root ; that each such group can be arranged in a number of 
sub-groups, the extent and the number of which are determined 
by the elementary divisors connected with the root ; and that the 
aggregate of the various groups of integrals, associated with the 
respective roots of the fundamental equation, constitutes a funda- 
mental system. 



Group of Integrals associated with a Multiple Root oe 
THE Fundamental Equation. 

Let K denote any such root of multiplicity a; and let the 
elementary divisors of A (8) in its determinantal form be 

{«-«)'—, (e-«)"-'-. .... («-«)'--"•-, («-«)'-■; 

then the minors of order r (and consequently of degree to — t in 
the coefficients of ^) are the earliest in increasing order which do 
not all vanish when d — k. Consequently, in the set of equations 

T of them are linearly dependent upon the rest; hence taking 
m — r which are independent, we can express m — t of the con- 
stants p linearly in terms of the other t, which thus remain 
arbitrary. Let the latter be pi, ..., p^; then the integral, given 

by 

U = p,W, +p2tVi +--. + pm1«m. 

becomes 

u = p,W, + p^W,+ ...+p,W„ 
where 

Tfi = Wi-l-A:,+i,|W^+i + ... + k„,iW,^ 

H^2 = Ws + /;,+,, aWr+i + . ■ . + h>i,2^n 
W, = iVr + &,+,,TWr+. + ■ ■ ■ + ^m.rW™ 

and the determinate constants k are given by 

Pr+i = h+,^,p, + k,+,^^p^ + ... + k^+j^. 
pr-n = h+ii,ipi + kri-i,sP^+ ■•■ +k,+2, 

Pm =/l^™,iPi +^m,ap2 + ■-- +4»,ti 



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54 ELEMENTARY [19. 

being the expressions for the m — r quantities p in terms of the 
T quantities p which remain arbitrary. 

Evidently each of the quantities W is an integral of the 
equation : and they have thc! property 

w;=kW,. 

for r = l, ,,., T. Moreover, they are linearly independent; any 
non- evanescent i-eiation of the form 

would lead to a relation between w,, ..,,Wm which would be homo- 
geneous, linear, and non-evanescent, a possibility excluded by the 
fact that Wi, ,.., if™ constitute a fiindamental system. 

The only case, in which t = o-, occurs when the indices cr — o-, , 
CTi — (Tj, ..., tTr-i of the elementary divisors are each unity. In 
that case, we have obtained a set of integrals, in number equal to 
the multiplicity of the root. 

20, We shall therefore assume that t < cr ; and we then use 
the integrals W,, ..., W, to modify the original fundamental 
system w,, ..., w^, substituting them for Wi, ..., w^. When the 
variable z describes a simple closed contour round a, the effect 
upon the elements of the modified system is to change them into 
IT/, W^', ..., W,', «/',+„ ..., w'n, where 



W,f^ 



cW^, 



■...+8.,«^ 



for r — 1, ..., T, and s = t+ 1, ..., m. The fundamental equation 
derived from this system for the singnlaiity a is 

.4(n) = o, 

where 



K-a, ,. 


.. 







,.,. 





, «-n,. 


. 







.... 





0.0,. 


., K-a 







.... 





At.... A„.... 


. A+.„ 


fi,^ 


.„+. 


-n, A+,„,„... 


I3,„,,. 


/3„ , /3„ ,. 


., /3.„ 


A. 


.« 


, /3»,,+, ,.., 


&„-f 



.(«-n)'A,<ii), 



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20.] 
where 



As « is a root of A (U) of multiplicity <t, it is root of Ai (Xl) of 
multiplicity er — T ; and a question arises as to the elementary 
divisors of A^ (fi) associated with k. 

The elementary divisors of A-^ (fl), which are powers of k — Xi, 
are 

(n-«)'-.-", (n-«r-'.-i, (n -«)•—.->, ... 

being, in each instance, of index less by unity than those of 
A (H). This result, which is due to Casorati*, follows from the 
property that .4,(0) is divisible by (li — «)"-'; its iirst minora 
are divisible by (ii — «)">"''-" and not simultaneously by any 
higher power; its second minors are divisible by (fi — k)"'"''"^' 
and not simultaneously by any higher power ; and so on. 

Thi« property, that all the minors of A^ (Si) of order fi ai'e fliviaible by 
(jL. 11) '' ' ' and not isimultancously by any higher power, can be proved 
a follow t 

Aiy mmoi of order ^ of ^ (Q) must contain at least ra—r-ii of the last 
m T columns let it contain m-r—ix+a of these columns, where a can 
range from to /i. It then miiat contain r-o of the first r coJumne. 
^im larly it must contain at least m-r— it of the last nt — r rows : let it 
conta n r /i + a' of these rows, where a' can range from to ^. It then 

must cu tai r — q' of the first t rows. The minor may be identically zero : 
if not then ow ng to the early columns and early rows that are retained, it is 
livisible by (k 2^"", and possibly by a higher power of k — a. Conse- 
quentlj ome imong these minors are espreasible as the product of {« -fi)'"'' 
by a Imear combination of minora of Ai (Q) which are of order /i ; the coefii- 
eienta in the combination are composed of the constants, which occur in the 
first T — ^ columns and the last m-r rows, and thus are independent of O. 
But a minor of order it ol A (Q) is not necessarily divisible by a powei' of 
K - fi with an index higher than <t ; thus 

(i!-fi)V. polynomial in fi=(s-fi)'""'''.sum of minors of ^, (n). 
It therefore follows that the power of « - Si common to all those minors of 
A, {a) is of indei not higher than o- -(t — ^^). 

' Comptes Reitdus, t. xcii (I88I). p. 177. 

+ Heffter, MMeitmm in die Theorie der Uitearen DiJfsrenUulyleichvngen. 
pp. 350—256. 



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56 ELEMENTARY DIVISORS OF THE [20. 

Kext, we know that thore axo some minora of the original A (ii) of order t, 
which do not vanish when JJ = k and which therefore are not divisible by 
« — Q. Clearly they cannot contain any of the first t rows in A (Si) ; and thus 
they miat ho composed of sets of to — t columnB selected among the last 
m - r vows. Take the minors of order )i of any one of these non-vanishing 
determinants, their number being N^, where 






= ^; 



and denote these minora by 

ifi (/ A=l, ...,N), 

tl togors /adiurrei Iftttl 11 terat on of i et of ^ columns 
a d a set f /i rowa o t of the non an sh j, deter ant of order m - t. 
Let ft Ije the comj leme tarj of Y,^ n ta w determ na t 

Non take tl e nunor of 4 (Q) nh ch ire of o de /t the number ia iV', 
a d thej n aj be denoted by o j f o ^ = 1 \ w tl tl e o significance 

in the integers as for M/j^. Construct an expression 

say, where J^ is a determinant of order m-r. Then either (i), J^ vanishes 
identically, owing to identities of rows or columns ; or (ii), J^ ia equal to 
+ A^(il) and therefore is divisible by (^-fl)"'"'', that is, certainly divisible 
by («-n)°"^ "('■-''*, for (§16) wo have 

T — (T'l > (T^ — o-^ ^ . . . ^ u^ _ 2 5^ 1 ; 
or (iii) /ft, when bordered by r — fi of the first rows, and tho first columns in 
A (11), is a minor of order ^ oi A (JJ) and ia therefore divisible by (k— !J)''i', so 
that the equivalence of tho two espressiona for the minor of A (Q) gives 

(k — il)'^f . polynomial in a = (K- ilY~''.Jh, 
and therefore J^ is divisible by (jt-n)V~t''~'''. Tt thus follows that J,, is 
divisible by (K-fl)'^» "*''"'*', in every case when it is not aero: and this 
holds for all values of /(. Taking then 

rai,aii + n(jja(g + ... + mjwai,v=/i, 
for /i=i, ..., A' and for one particular value of i, we have a series of A^ linear 
eijuations in the quantities a;,, ..., 0;^-. The determinant of their coefficients 
is a power of the non- vanishing detemiinant of order to-t, for it ia a 
determinant of all its minors of one order : and therefore it doea not vanish. 
Hence, so far ispoRe'a t k Jl are conce ed e h of the mnoB a, a v 

is a linea co h n t of / ■/> <kll of 1 e e are d s ble by 

(k-Q)"«" '' d d he ef -e h of the m o ■s a. a jf s «rta nly 

divisible by h t power The esult 1 olds lo ea<;l t the val es of 

It has eea oe t t e i we o k a om no to 11 tl ese n ors of 

A^ (Q), has a n 1 ot greater ha a /i omb n^ tl e rea Its, ve 

infer that the highest power of k - Q, common to all the minors of A^ l_il) of 
order /i, has its index equal to ir - (r - p). 



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21.] FUNDAMENTAL EQUATION 57 

21, The indices of the elementary divisors of A^ {11) at-e 

ff-o-,-1, <r,-o-2-l, a,-(T,~l, ...; 

let there be t' of them, where t'$ t, so that the last t — t' of the 

indices of those of A (il) are equal to unity, on account of the 

property 

o- - 0-, ^ 0-1 - o-s ^ o-a - o-a > - . . 3^ T,-, > 1. 

Then the minors of .^i of order t' (and consequently of degree 
TO — T — t' in the coefficients of Ai) are the earliest in successively 
increasing order, which do not all vanish when il = k; conse- 
quently, in the set of equations 

Pl'0r.r+, + P-l^r,,^^ + ■ ■■ + pV, /3,,„, = Kp/, (r = r + 1, . , ., to), 

t' ol' them are linearly dependent upon the rest. Hence taking 
in — T — t' of the equations which are independent, we can express 
m — T — T of the constants p' in terms of the other t', which thus 
remain arbitrary and which may be taken to be p,', . . ., p'^. 
Now take an integral 

v = pM+. + ... + p\n-.w..> 
and substitute for the various coefficients p' in terms of p,', ..., pV- 
The integral becomes 

V = p! TF"„ + p/ ^fj^ + - . . + p',- t^lr' ■ 

where, writing \ — t + t\ we have 

Wir = W^+r + k+i,rVlk+i + . . - + Un,rWm, 

forr=l, ...,t'; and the determinate constants i are given by 

Pr'+s = /j,+B,ip,'+ ... + ^J.+s, r'pV, 

for 8 = 1, .... m — X, being the expressions of the constants p' in 
terms of ,0,', .... p',-. 

Clearly ea<;h of the quantities W-a, ^a, ■■■, W-^' is an integral 
of the equation. Moreover, they are linearly independent of one 
another and of W], ..., W,\ for any non-evanescent linear relation 
of the form 

F,W, + ... + F,W,^F:W,, + ... + fVTf.y = 
would lead, after substitution for W-^. ..., Wj, W"„, ..., Wi,- in 
terms of the original fundamental system Wj, ...,Wm, to a non- 
evanescont homogeneous linear relation among the members of 
that system — a possibility that is excluded. 



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58 SUB-GROUl'S OF [21. 

As regards the effect, which is caused upon each of these 
newly obtained integrals by the description of a simple contour 
round the singularity, we have 

Tf,,' = W\^, + k^,.rVj\+, + ... + ln,,-y>«' 
= lcW,r+ V,., 

where V,. denotes a homogeneous linear combination of TTi, ..., 
W,. Now no one of the quantities Vy can be evanescent, nor 
can any linear combination of the form 
7iFi + ... +7t'F,' 
be evanescent : for in the former case, we should have 

and in the latter 

(7, w„. + . . . + 7,' w„.y = « {-y, If „ + , . . + y,, r,.o- 

As Wir and y,W„ + ... +'y.,'W,r- in the respective cases are linearly 
independent of W^, ..., W,, we should thus have a new integral 
of the same type as Wi, ..., Wr', and then, instead of having some 
of the minors of order t in A (H) different from zero when H — k, 
all of them of that order would be zero, and we should only be 
able to declare that some of order t + 1 are different from zero : 
in other words, the number of elementary divisors of A (H) would 
be T + 1 instead of t. The quantities V^, ..., V^ are thus linearly 
equivalent to t' of the quantities Wj, ..., W,, say to Wi, ..., TT^'; 
hence constructing the linear combinations of V,, ..., V^i which 
are equal to TT,, ,.., Wj' respectively, and denoting by w,,, ..., 
Wir' the linear combinations of W^, .... W^^' with the same coefGci- 
ente as occur in these combinations of V,, ..., V,', we have a set 
of t' integrals «?„, .... Wi^, such that 

tVir = KWir+ Wr, (r = 1, ..., t'). 

These integrals are linearly independent of one another, and also 
of Wj,..., Wr, before obtained. They constitute the aggregate 
of linearly independent integrals of this type ; for if there were 
another linearly independent of them, it would imply that A^ (li) 
had t' + 1 elementary divisors instead of only t'. 

As regards the two sets of integrals already obtained, it may 
be noted, (i), that the set TFi, ..., W^ can be linearly combined 
among themselves, without affecting the characteristic equation 
WJ^kW,.: 



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21.] INTEGRALS 59 

(ii), that to aach integral of the set Wn, .... Wi,< there may be added 
any linear combination of the integrals of the set Wj_, ..., W^, 
without affecting the characteristic equation 

If the index of each of the elementary divisors of Ai(il) is 
unity, then t'—ct^t, so that the number r + r of integrals 
obtained is then equal to <t, the multiplicity of the root of 
A (Xi) = in question. In every other case, ■/ + t< a: 

22. When t' + t < tr, so that t' is less than the degree of 
^i(il), we use the integrals w„, ...,w,,/ to modify the funda- 
mental system W^, ..., W^, w^^j, ...,Wm, substituting them for 
w,+,, .... «!,+/ in that system. When the variable z describes a 
simple closed contour round a, the effect upon the elements of the 
modified fundamental system is to change them into IJV, .... W,', 
Wii, ..., w'it', w\^i. ..., wj,, where t ^-t —\, and 

w; = kW„ 

w/-7ftF,-l-... + 7„V. 

+ 7t,T+iW„ + ... + 7tAWi,' + 7(,A+,WA+,+ ... +7i,mW,„, 
for r = 1, ..., t; s = 1, ..., t'; f = \+ 1, ..., m. The fundamental 
equation derived through this system is 

A (fl) = {« - Iiy+''^, (11) = 0, 
where 

J.j(Il)=l 7x+,,*+, -ft, 7A+i,*+si ■■-. 7*+i,™ I- 



17™,''+! . 7™,^+^ . ■■■, 7m,m-ft| 

Also * is of a root of A^in,) of multiplicity <t — t--t'. By a 
further application of the proposition (§ 20) connecting the 
elementary divisors of A (il) and -4, (H), the indices of the 
elementary divisors of ^^{n), which are powers of k — CI, are 
seen to he 

ff — (7] — 2, (7j — (Tj — 2, (T^ " o^s — 2, .... 

say t" in number. 

The procedure from the equation A^ (fl) = to the corre- 
sponding sub-group of integrals is similar to that adopted in the 
case of the equation .A, (H) — ; and the conclusion is that there 



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60 COMPOSITION OF A GROUP [22. 

exists a sub-group of r" integrals w^i, iCs,, ..., Wj^", characterised 
by the equations 

for ( = 1,2, ...,t". 

And so on, for the sub-groups in succession. Combining these 
results, we have the theorem* ; 

When a root k of the fmidammtal equation A (fi) = is of 
■multiplicity/ <r, and when the elementary divisors of A (12) associated 
vjith that root are 

a group of a lineai ly mdtpendpnt iiitegt als is associated with that 
root : this group constat-' if a mimher /r — a-i of sub-groups, which 
satify the equatioii', 

Wf = KWr, for r=l, T, 

Wa = KW^ + w,t, for t = l, ..., t", 
and so on. The integer r is the number of elementary divisors of 
A (ii) ; t' is the number of those divisors with an index greater 
than unity ; r" is the number- of those divisors with an index greater 
than two ; and so on. 

The group of <r integrals, and in~ a- other integrals, all 
linearly independent of one another, make up a fundamental 
system : tlie m. — a other integrals being associated with the 
m — (T roots of ^(li)=0 other than D^—k. When these roots are 
taken in turn, wc have a single integral associated with each 
simple root, and a group of integrals of the preceding type asso- 
ciated with each multiple root, the number in the group being 
equal to the order of multiplicity of the root. We thus have a 
system of integrals of the original differential equation distributed 
among the roots of the fun<lamental equation associated with the 

" That pait ot the theoiem which establialiei the e^isteiice of the group of 
inteitralB ata ttated with a multiple loot is due to Fuoh'i Cielle t lwi (18fl6), 
p. lifb but the initial eipiession ti'O 'o 'he members of the group was much 
moie complicated The part which arranges the group m sub gioups each with 
its own chaioeteristic eijuafion is due to Hamburgei CrdU t Litvt (1873) 
p. 121 he takes it in an aiiangement \vhieh will be found in the next seotiou 
The association of the sub-groupR with the elementaiy divii^oic of i \Q] i& due to 
Canorati Compte' Kemiits, t sen (1881) p 177 



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22,] OF INTEGRALS 61 

singularity : that the system is fundamental is manifest from the 
facts, that the initial system was fundamental, and. that all modi- 
fications introduced have been such as to leave it fundamental. 

Ex. 1. Two independent int^rala of the equation 



Hence when tho variable describes a sirajilc closed contour round the origin 
it) the positive direction, we have 



and therefore the fundamental equation belonging to the origin (which is a 
singularity of the equation) ia 

I -\~e, 1=0, 
I -2jrt , -\^e ■: 

that is, it is 

(fl+i)'.o. 

Similarly, two independent integrals of the equation 
„dho , dw „ 
are given by 

Hence aftei' a simple closed contour round the origin, we have 

where n is e^" ; the fundamental equation belonging to the origin is 

: -l_S, 1 = 0, 

I , a-6 ' 
thatia, 

Ex. 2. Consfruct the linear differential equation of the third order, 
having 

for three b early ndepe del t ntegra!'' btain the fundamental equation 
apperta n ng to the r g as a a gular ty ; and from the form of the 
diflere tal eq at nn ver ty P" car^a theorem (§14) that the product of 
the three r t's of th a t ndame tal e'j t n ia unity. 



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62 hamburger's [23, 

Hamburger's Resolution of a Group of Integrals into 
SuB-oROUpa 
23. In the case when the roots of the fundamental equation 
are all distinct from one another, the general analytical character 
of each of the integrals of the fundamental system in the vicinity 
of the singularity has been obtained (| 18). We proceed to the 
corresponding investigation of the general analytical character 
of the group of integrals in the vicinity of the singularity, when 
the group is associated with a multiple root of the fundamental 
equation. 

We have seen that the group of iiuearly independent integrals 
can be arranged in sub-groups of the form 
W„ W.„ .... W, ;. 



the members of eaeh sub-group being aiTanged in a line and 
satisfying an equation characteristic of the line. Let these be 
rearranged in the form* 

1^1, Wn, Wu, w^, ... 

Tfa, W]s, Ws3, Wjs, ■-■ 



each of the integrals in the new line satisfies an equation, and the 
set of characteristic equations for any line is, in sequence, the 
same as for any other line, so far as the members extend. When 
any such line is taken in the form 



where the integer fi changes from line to line, the set of the 
characteristic equations is 



* These are Hamburger's sub-gconps; see note, p. 60. Their number is equal 
>o the nwnbei of elemental; diviBois of A (Q) connected with the multiple root. 



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23.] 




SUB-GROUPS 


Let 

we have 
and therefore 


[(^ 


'i-iria = log K 
-»)•]■ = «(.- 



[% (z — a)~"]' = «! {z — «)~°. 
Thus «, {z — «)~" is unaltered by the description of a simple closed 
contour round a; it therefore is uniform in the vicinity of a, but 
it cannot be declared holomorphic in that vicinity, for a might be 
a pole or an essential singalarity of u^ {z — a)"". Denoting this 
uniform function of 2 — « by -^i, we have 
u, = {z-aY^^. 
To obtain expressions for the other integrals, Hamburger* 
proceeds as follows. Introduce the function L, defined by the 
relation 

i.ilog(.-,.), 
then, after the description of a simple contour, we have 

L' = L + l. 
We consider an expression 

F(i)=j'=^.,+(''-^)t,-.i+('';V"-''+- 

where 

\ T ) (^-1-r)! r!' 
and the functions -^i,...,-^^ are uniform functions of z — a. 
Then if, for all values of n, we take 
7/^^, = (z-ay^-APF, 
where the symbolical operator A is defined by the relation 

^F^F{L + 1) - F{L} = F'-F, 
we have 

= (3 - a)''«''+^ (A"F+ A^-^'F) 
" Crdle, t. LJLXT! (1873), p. 122. 



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64 GENERIC FORM OF [23. 

holding for all values of n. These are the characteristic equations 
of the modified sub-group ; and therefore we can write 

with the above notations. This is Hambui'ger's functional form 
for the integrals, 

24. The integrals ii,, ..,, u^ are a linearly independent set 
out of the fundamental system ; and the system will remain 
fundamental if Mj, . . . , w^ are replaced by /^ other functions, linearly 
independent of one another and linearly equivalent to Mj, ,.., m„. 
A modification of this kind, leading to simpler expressions for the 
sub-group of integrals, can be obtained. In association with F, 
take a series of quantities, defined by the relations 
»', = ^„ 

f^ = 11^5 + 2-^.^L + y{r,L'', 

i^=«. = >^.+ (''->«-,i+(^2>.-.i^4-... 

Then we have 

AF=auV^_, + a,^v^_^ + a„v^^,+ ... +a,,^_,v,. 



where the constants a are n on- vanishing numbers, the exa«t 
expressions for which are not needed for the present purpose. 

Then (e - ayv, is a constant multiple of (ir - ayAi^-^F, that is, 
of u, ; and it therefore is an integral of the differential equation. 

By the last two of the above equations, (s — aYv^ is a linear 
combination of (£— a)°A''~'F and (z — ayA''~'F, that is, it is a 
linear combination of Jia and Mi ; it therefore is an integral of the 
differential equation. 

By the last three of the above equations, (2 — ayv^ is a linear 
combination of (e - afA'^-'F, {z - afA^-'^F, {z - a)''A»-'^; that is. 



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24.] A GROUP OF INTEGRALS 65 

it is a linear combinatioa of u,, Us, Mj ; it therefore is an integral 
of the differential equation. 

Proceeding in this way, we obtain /j. integrals of the form 

{^~ay,„ {z-aYv, (^-a)-«,. 

Moreover, these are linearly independent, and so are linearly 
equivalent to it,, ,.., m^; for, having regard to the expressions of 
AF, ..., A«~'f, we see at once that any homogeneous linear rela- 
tion among the quantities v-^, ...,% would imply a homogeneous 
linear relation among the quantities F, AF, ..., A''~^F, that is, 
among u^, ..., u^; and no such linear relation exists. Hence 
Hamburger's sub-group of integrals is equivalent to {and can be 
replaced by) the sub-group 

{.~o)-.„ (^-.)-., (^-o)-»„ 

Accordingly, we now can enunciate the following result as 
giving the general analytical expression of the group of integrals, 
associated with a multiple root « of the fundamental equation*:-— 

Wke}i a root k of the fundainental equation A{d)—Q is of 
multiplicity a, the group of <t iniegrals associated •with that root 
can he arranged in avh-groups ; the number of these sub-groups is 
equal to the number of elementary divisors of A($) which are 
powers of k — 6; the number of integrals in any sub-group is 
determined by means of the exponents of the elementary divisors; 
and a sub-group, which contains fj. integrals, is linearly equivalent 
to the /J. quantities 

(z-ay,,. {z-afv, (^-»)-.„ 

where 27ria = log k, and the ji. quantities y are of the form 

V, = yjr, + 2-^^L + f,L\ 



' This form of expresHion for the gronp of integrals appears to have been given 
first by Jiirgens, CreUe, t. Lxsx (1875). p. 154. See also a memoir by Fuchs, 
Berl. SitzangiUr., 1901, pp. 34—48. 



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66 SUB-GBOUP OF INTEGRALS AND [24. 

where L = „ .log(z — a), (^ ] denotes -. q r;~, . f"*"^ 

Stti ^' ' \ r j . {p.-\—r)\ t\ 

the jj. quantities i/^j, ;.., ■^^ are uniform (but not necessarily 

holomorphic) functions of z — a in the vicinity of the singularity. 



Dll'FEBENTIAL EqL'ATION OV LoWEE ObDEK SATISFIED BY 

A Sub-Group of Integrals. 

25. The preceding form of the integrals in each sub-group of 
a group, associated with a multiple root of the fundamental 
equation, has been inferred on the supposition that the coefficients 
of the linear equation are uniform functions. 

It will be noticed that the coefficient of the highest power of 
L in each of the members of the sub-group is the same, being an 
integral of the equation, — a result which is a special case of a 
more general theorem. Moreover, it is of course possible to verify 
that each member of the sub-group satisfies the differential 
equation ; and it happens that the kind of analysis subsidiary to 
this purpose leads to the more genera! theorem above indicated, 
as well as to a result of importance which will be useful in the 
subsequent discussion of the reducibility of a given equation. We 
proceed to establish the following theorem* which is of the nature 
of a converse to the theorem just established : 

If an expression for a quantity u be given in the form 
U = tj}n + <f>n-,L + ^„^L^+ ... + ^ai"-' + ii>iL"-\ 

where L = ^ — . log (z — a), and each of the quantities is of the 

form 

(^ = (^ — a)' . uniform function of z~a, 

a being a constant, then u satisfies a homogeneous linear differential 
equation of order n, the coefficients of which are functions of z 
uniform in the vicinity ofz = a; moreover, 
du ^ 3"-'M 

dL' dL" '"' 9i»-' 
are integrals of the same equation and, taken together with u, they 
constitute a fundamental system for the equation. 

" Fiiclis. in the memoiv quoted on the pceeedins page. 



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25.] ITS DIFFERENTIAL EQUATION 67 

(It is clear that „,^^^ is a numerical multiple of ij>,, and that 

the coefficient of the highest power of L in each of the announced 
integrals is, save as to a numerical constant, the same for all ; it 
is a multiple of 0i, which is an integral of the equation.) 

It is convenient to make a alight modification in the form of u ; 
we take 

■■■+("i')+'^""+ *'-'''"■ 

where 

so that the character of the functions i^ and their form (except 
as to a mere numerical constant) are the same as those of the 
functions i^. Further, no change, either in the property that 

dii_ dhi 

are integrals of the equation or in the property that, taken 
together with u, they constitute a fundamental system, will be 
caused if they are multiplied by constants : so that, if the theorem 
can be established for -n,, ..., m„_j, where 

1 3"-'« , 
1 ! 8"-i, , , , r 






y^,L'^^ + ir,L«-' 



the theorem holds for the quantities as given in the enunciation 
of the theorem. 



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68 SET OF LINEARLY [26. 

26. Merely in order to abbreviate the analysis, we take m = 4 ; 
with the above forms, it will be found that the aaalysis for any 
particular caee such as n = 4 is easily amplified into the analysis 
for the general case. Accordingly, we deal with quantities w, u,, 
u-i, u?,, where 

M = i|^j + S-^-jL -I- Z-^M + ir^L", 

If u can be an integral of a linear equation of the fourth order 
with coefficients that are uniform functions oi' s — a in the vicinity 
of a, let the equation be 

dz' ds^ dz^ ds 
Let the variable z describe a simple contour round a ; this leaves 
the differential equation {if it exists) unaltered, and so the new 
form of u is an integral, say u, where 

u'=Kf,+ BKf,{L + 1) + dKf,{L + ly + K^p■,{L + If, 
where k is the factor comnjon to all the functions ifr after the 
description of the circuit. As u and u' are integrals of a homo- 
geneous linear equation, so also is 

Hence v also is an integral, and it is given by 
v' = B {«V^a + 2«i^, (i + 1) + K^i (£ + If} 

+ 3 [Kf, + Kf, (L + 1)1 + K^, ■ 
and therefore 

w, = ^ [ - I'' — w I = Mj + M, , 
ntegral. Hence w' is also an integral, and it is given 



by 

and therefore 

is also an integral. 



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26.] INDEPENDENT INTEGRALS oS* 

Thus integrals are given by 

t, =■«!, 
■w-t, =«5, 

which proves one part of the theorem, viz, that u, u,, u^, Uj are 
simultaneous integrals of the linear equation if it exists. 

27. In order to estabhsh the property that w, Ui, u^, % con- 
stitute a fundamental system of the equation if it exists, a pre- 
liminaiy lemma will be useful; viz. if A, B, G, D be functions 
free from logarithms and if they be such that a simple closed 
contour round a restores their initial values, except as to a con- 
stant factor the same for all, then no identical relation of the 
kind 

aA + 0BL + yCD + BDL' = 

can exist, in which a, /3, y, S are constants different from aero. 
For let the simple contour be described any number, N, of times 
in succession ; and let / be the constant factor acquired by the 
functions A, B, 0, D after a single description of the simple 
contour. Then we should have the relation 

/"[aA + 0B{L + N) + yC{L + Nf + SD(L + N'f] = 0, 
and consequently the relation 

aA + 0B(L + N} + yG(L + Ny+?iD{L + Nf^O. 
valid for all integer values of JV". Consequently, the coefficients of 
the various powers of JV" must vanish : hence 

0= SD. 

= 38Z)i + yC, 

= 3SDi= + 27CZ -I- &B, 

0= hI)L'-\- yGL^ + l3BL + <xA, 

the last of which is the original postulated relation. From the 
first of these relations, it follows that 

S = 
then, from the second, that 

7 = 
then, fi-om the third, that 



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70 EQUATION SATISFIED BY [27. 

and so, from the original relation, that 

a = 0. 
The lemma is thus established. 

It may also be proved that, if A, B, C, D be functions free 
from iogarithma, and if they be such that a simple closed contour 
round a restores their initial values, except as to constant factors 
which are not the same for all, then no identical relation of the 
kind 

aA + ^BL + riCD + BDL^ = 

can exist, in which a, 8, 7, 2 are constants different from zero. 
The proof is left as an exercise. 

It is an immediate inference from the course of the lemma 
that no relation of the form 

a'u + ^Ma + Jli^ + B'Ui = 
can exist, in which a, ^', i, S' are constants different from zero ; 
for proceeding as before, it would require 
0= a'-.|r„ 

= 3a' ■f 3 + 2/3>5 + 7>i, 

= n'->^, + ^'--^s + i-if-i + ^'-^1, 

which clearly are satisfied oniy if a' = ^' = 7' = 8' = 0. Hence there 
is no homogeneous linear relation among the quantities m, Mi, Ws, 
Ms ; and they therefore constitute a fundamental system for the 
linear equation if it exists. 

28. If the equation exists, we must have 

and in the operator A, the functions P, Q, ii, S are to be uniform 
functions of z in the vicinity of a. Let Z denote any function of z 
with the same characteristic properties as ^], ^.j, if~a. "^4; then 
with such an operator A, we have 

A(^i) =iA2 +Z', 

A(2'i=) = Z=A.?+2L2' ■\-Z", 

A (^i') = i^AZ + ZL'Z' + %LZ" + Z"\ 



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28.] A HET OF IiNITGGRAJ.S 71 

where Z' , Z", Z'" are functions of the same characteristic properties 
as Z, that is, aa i/^j, i^^, -Jcj, i^^, and they are iree from logarithms. 
Now as A?i, = 0, we have 

A-.^. - 0. 
As A«,; — 0, we have 

ii|^,j + iA'^i + -l^-i' = 0, 
that is, 

As A«a = 0, we have 

A-f 3 + 2 (LAi^i 4- f /) + i^A--^, + 2i^/ + -./r," = 0, 
that is, by using the two preceding relations, 

As A((=0, wehave 
A->/rj + 3 (LAa/t, + ■^;) + S (Z^A-»/r, + 2if / + 1/^;') 

+ i^Ai/Tj + ZL^; + 3i/f," + -^1" = 0, 
that is, by using the throe preceding relations, 
Ai^^ 4 3i^/ + 3-f /' + i/r,"' = 0. 

Thus there are four equations ; each of them involves the coeffici- 
ents P, Q, -R, S linearly and not homogeneously. The required 
inferences will be obtained if the equations determine P, Q, R, S 
as functions of s, uniform, in the vicinity of a. 

Now each of the functions -yjr is such that {z — ay-'yjr is a 
uniform function of 2 - a in the vicinity of a ; accordingly, let 

(z-a)--^f^=0^, (/t=l, 2,3,4), 

where each of the ^'s denotes a uniform function. Substituting 
(z — ay$i^ for -^1^ in each of the four equations, the factor (s — a)' 
can be removed after the dififerential operations have been per- 
formed ; and then all the coefficients of P, Q, R, S, and the term 
independent of them, are uniform functions of ^ in the vicinity of 
a. Solving these four equations of the first degree for P, Q, M, S, 
we obtain expressions for them as uniform functions of s — a in 
the vicinity of a. (In general, this point is a singularity for each 
of the expressions.) It follows that, for these values of P, Q, Jf, S, 
the four quantities Wj, u^, v^, u are integrals of the linear differen- 
tial equation of the fourth order. 



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72 PROPERTIES OF A SUB-GKOUP [28. 

As already remarked, similar analysis leads to the establish- 
ment of the result for the general case ; and thus the theorem is 
proved. 

Corollary I. It is ao obvious inference from the preceding 
theorem that, when a group of integrals is associated with a 
multiple root of the fundamental equation, any (Hamburger) 
aub-gi'OTip, containing (say) u of the integrals, is a fundamental 
system of a linear equation of order n with uniform coefficients. 
Further, it is at once inferred that the n' members of that sub- 
group, which contain the lowest powers of the logarithm, constitute 
a fundamental system for a linear equation of order n' with uniforiu 
coefficients. 

Corollary II. Similarly it may be established that one 
(Hamburger) sub-group containing n integrals, and another sub- 
group containing p integrals, constitute together a fundamental 
system for a linear equation of order w-l-^ with uniform coefficients. 
And so on, fur combinations of the sub-groups generally. 



Jix. Prove that if the linear equation in w has a sab-group of n iiitegi'als 
which, ill the vicinity of a singularity a, have the form 

W3 = ./'3 + 21;'^Z;-i-^/',Z^ 



where 2ii2X = li>g(a-m), and oaflh of the functions^ is sueli tbat (s-a) "^ is 
uniform, where e^'^ is a multiple root of the fundamental equation with 
which the sub-group of integrals is associated, then if the linear equation for 
V he eonatructed, where 

that iineiir equation has a corresponding auh-group of n 



- 1 integrals of the 






where the functions ip are of tlie same character aa the functions tJt. 



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CHAPTER III. 

Regular Integrals; Equation having all its Integrals 
rbqular near a singularity. 

29. The general character of a fundamentai system of 
integrals in the vicinity of a singularity has now been ascertained. 
For this purpose, the main property of the linear equation which 
has been used, is that a is a, singularity of the uniform coefficients ; 
the precise nature of the singularity has not entered into the 
discussion. On the other hand, the functions <p which occur in 
the integrals are merely uniform in the vicinity of a : no know- 
ledge as to the nature of the point a in relation to these functions 
has been derived, so that it might be an ordinary point, or a 
pole, or an essentia! singularity. Moreover, the index r in the 
expressions for the integrals is not definite ; being equal to 

s— ■ log 6, it can have any one of an unlimited number of values 

differing from one another by integers. Hence, merely by changing 
r into one of the permissible alternatives, the character of a for 
the changed functions ^ may be altered, if originally a were 
either an ordinary point or a pole : that character would not be 
altered, if a originally were an essential singularity. 

It is obvious that the character of a for the integral is bound 
up with the nature of a as a singularity of the differential 
equation, each of them affecting, and possibly determining, the 
other. Accordingly, we proceed to the consideration of those 
linear equations of order m such that no singularity of the 
equation can be an essential singularity of any of the functions 
1^, which occur in the expression of the integrals in its vicinity. 
In this case, the functions (f>, which are uniform in the vicinity of 



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74 REGULAR INTEGRALS [29. 

a and therefore, by Laurent's theorem, can be expanded in a series 
of pasitive and negative integral powers of ? — a converging within 
an annulus round a, will at the utmost contain only a limited 
number of negative powers. To render r definite, we absorb all 
these negative indices into r by selecting that one among its 
values which makes the function ^ in an expression 

{z-af,!, 
iiuite (but not aero) when e= «. 
An integral of the form 

if = (3 - ay [<^, + <^, log (3 - a) + . . . + <^, |log (2 - a)Y\ 
where 0^, 0i, ..., 0, are uniform functions having the point a 
either an ordinai-y point or a zero, is called* regular near a. 
When a value of r is chosen, such that {z — a)~''u is not zero 
and (if infinite) is only logarithmicaiiy infinite like 
Co + c, log (2 -«) + ... +c„ [log (s -»)}", 
the integral is said to belong to the index (or exponent) r: the 
coefficients c being constants and not all of them zero. Similarly, 
when the singularity a is at infinity, and there is an integral 



|^^, + t.'''g^+-+t-(logJ)j. 



where ■^„, if-i, ..., i^, arc uniform functions having s = i» for an 
ordinary point or a zero, the integral is said to belong to the index 
or exponent p. 

It will be possible later to consider one class of integrals that 
do not answer to this definition of regularity : but it is clear that 
regular integrals, as a class, are the simplest class of integrals, and 
that the first attempt at obtaining integrals would be directed 
towards the regular integrals, if any. Accordingly, we proceed to 
consider the characteristics of linear differential equations which 
possess regular integrals : and in the first place, we shall consider 
equations all of whose integrals are regular in the vicinity of one 
of its singularities in order to determine the form of equation in 
that n tj 

A Th m I ) P T name for a 



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AND THEIR EXPONENTS 



As subsidiary to the investigation, one or two simple properties, associated 
with the indices to which the fuQctioiM helong, will first be proved. 

If a regidar fimction u (in the present sense of the term) hdong to an 
index r avd another v to an index s, tken M-i-w belongs to the index r — s: as is 
obvious from the definition. 

If a regular function u belong to am index r, t/ien -j- belongs to Ike index 
r—^, To prove this, let 

J.(.-«)—[_M>-*.+(.-«)*;)<loi5(,-,.)}-+.*.pog (.-»))— ], 

SO tiat -T- can onlj belong to the index r - 1, if some at least of the coefficients 
of powers of log{2 — w) are different fTOm zero when z = a. These coefficients 



when s = a is substituted: they cannot all vanish, for then </)„, ip,, ..., ^ 
would vanish when s = a, so that o,, c,, ..., i!^ would all be zero, and then u 
would not belong to the index r. Thus -r- belongs to the indes r — 1. 

There is one slight exception, viz. when « is uniform and the index r is 
zero ; then -y- is also uniform, and it may even vanish when 2 = a ; so that, 

if •« were said to beloi^ to the index 0, -5- could be said to helong to an index 
not less than 0. 



Form of the Differential Equation when all the 
Integrals are regular near a Singularity. 

30. As a first step towards the determination of the form of 
a differential equation that has all its integrals regular, we shall 
obtain the index to which the determinant of a fundamental 
system belongs. Let the system be w,, w^, ..., w„: and let the 



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76 




Ft 


>RM OF DIFFERENTIAL 


EQUATION 


[30. 


indices of the members be r„ r^ 
the determinant in the form 


...., r 


■,„ respectively. 


We take 








Cw/^tf,'"-' 


'«,"-. 






ofS 


3^2, 


where G i 


s a constant. 










The 


: quantity 


% is a solution 


of an 


equation, a fundamental 



system for which is given by 

It is clear that, if w,, w^, ... are all free from logarithms, then 
%, Vs, ... are also free from them. If hovrever there bo a group or 
a sub-group of integrals associated with a repeated root of the 
fundamental equation, we may take (§ 23) 

Wi' = Ow,, w/ = Wi + 6w^, 
so that 

, _ d /w.J\ _ ^ A ^\ 
' dz \W|7 dz\8 wj " 
thus 1), is uniform and therefore free from logarithms. Similarly, 
Ui and all the quantities used in the special form of the determ- 
inant are free from logarithms. 

The indices to which Vi, %, ... respectively belong are 
ra — r, — 1, r, — ri ~ 1, r, — r^ — 1, . . . 

unless it should happen that, for instance, r2 = rj. In that case, 
we replace Wa by w^ + av^i, choosing a so as to make the new 
integral belong to an index higher tlian r^ or r, : this change will 
be supposed made in each case where it is required. 

Again, the quantity m, is a solution of an equation, a funda- 
mental system for which is given by 

' dz \Vi' ' ' ds \vj ' 
The index to which % belongs is 

r,-r,-l-(r,-r,-l)-l, =.-.-r.-l, 
and so for u^, ... ; that is, their indices are 

'*a — ^a — !> r^ — r^—l,.... 
And so on, down the series. 



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30.] HAVING REGULAR INTEGRALS 

Hence the index to which 



= w, + (wi-l)(r,-r,-l) + (m-2){r,-r,-l) + .,. 

... + l(r.-r.-.-l) 

so that, denoting the determinant of the fundamental system by 
A {z) as in § 9, it follows that, in the vicinity of the singularity a, 
we have 

A (s) = (2 - a)'-'+^=+-+^'»-!™'"'-^i a (s - a), 

where Ji is a holomorphic function of its argument in that 
immediate vicinity, and does not vanish at a. 

31. This result enables us to infer the form of the differential 
equation in the vicinity of the singularity a. Manifestly, the 
equation is 

d'^w d'"-'w d™~^w 



where A is the determinant of the m integrals in the fundamental 
system, and A, is the determinant that is obtained from A on 

d'^-'Wg . ^, , d™w» 

. -, , for s = 1, . . . , in, by the column -, — , 

dz"'-" •' dz™ 

fors = l, ..., m. 

Now consider a simple closed path round a. After it has been 
described, A and A^ resume their initial values multiplied by the 
same constant factor, which is the non-vanishing determinant of 
the coefficients a (§ 13) in the expressions for the transformed 
integrals ; thus p^ is uniform for the circuit. Hence, when the 
expressions for the regular integrals are substituted in A and A„, 
all the terms involving powers of log {z — a) disappear. Moreover, 
A belongs to the index 

n+...+r™-Jm(m-l); 

and so far as concerns the index to which A^ belongs, it contains a 

column of derivatives of order k, =m — {i)i — k), higher than the 

corresponding column in A, so that A„ belongs to the index 

n + ...+r,.-im(m-l)-«. 



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78 METHOD OF FEOBENIUS [31. 

Hence p^ belongs to the index — k and therefore, in the immediate 
vicinity of a. the form of j), is given by 

where, at a and in tiie immediate vicinity of a, the function 
Pk{z — a) is a holomorphic function which, in the most general 
instance, does not vanish when z = a, though it may do so in 
special instances. As this result holds for k = 1, ..., m, we con- 
elude that, when a homogeneous linear differential equation of 
order m has all its integrals regular in the vicinity of a singularity 
a, ike equation is of the form, 



in that vicinity, where P,, Pj, ..., P^ are holomorphic functions of 
z — ain a region round a that encloses no other singularity of the 
equation. 

Construction of Regular Integrals, ey the Method 
OF Fbobenius. 

32. The argument establishing this result, which is due to 
Fuchs*, is somewhat general, being directed mainly to the 
deduction of the uniform meromorphic character of the coefficients 
of the derivatives of w in the equation. No account is taken of 
the constants in the integrals i and it is conceivable that they 
might require the esistence of relations among the constants in 
the functions P,, ..., P,„, Hence for this reason alone, even if for 
no other, the converse of the above proposition cannot be assumed 
without an independent investigation. The conditions, which 
have been shewn to apply to the form of the equation, are 
necessary for the converse: their sufficiency has to be discussed. 
Accordingly, we now consider the integrals of the equation in 
the vicinity of the singularity!. 

Denoting the singularity by a, we write 

z-a = x. PAz-a) = pr{x)=p., (r = l, ...,m); 

* CrelU, t. Lsvt (1866), p, 146. 

t The following method ia due to Frobciiius ; leferenMS will be given later. 



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32.] FOR REGULAR INTEGRALS 79 

SO ihat the equation can be taken in the form 

valiii in the vicinity of a: = 0. 

If regular integrals exist in this vicinity, they are of the form 
indicated in §§ 18, 24, the simplest of them being of the form 

wf = a^ 2 g„x° = 2 g^xi''^" 

say ; should this be an integral, it must satisfy tlie equation 
identically. We have 

D«-= |,7(<7-1)..,(» -m+ !)-»(„ -!)...(» -m+2)j),-.. . 

say. Here,y(i«, tr) is a holomorphio function of x in the vicinity of 
the iC-origin and is a polynomial of degree m in (j, the coefficient of 
(7™ being unity i so that, if it be arranged as a power-series in x, 
we have 

f(x.,).f,{„) + ^fA,) + x-M,) + .... 

where /„ (tr) is a polynomial in ff of degree m, and /,{o), f^ (/r), . . . 
are polynomials in cr of degree not higher than m—1. Then 

= !/.«'*'/('»- P + ») 

= iai'+-fj,/.(p + ») + y,_,/,(p + «-l) + ... + ?./,«). 

If the postulated expression for w is to satisfy the equation, the 
coetBcienta of the various powers of x on the right-hand side must 
vanish : hence 

o-i/./.W, 

-»/.((>) +?./.(p + i), 

O-S./.W + »/,((> + 1)+S./.(P + 2), 
and so on. These equations shew that the values of p, which arc 
to be considered, are the roots of the algebraical equation 

fM-o 



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80 CONSTRUCTION OF [32. 

of degree m in p: and that, for each such value of ^, 



s. = -, 



_!h_ 



7.(P + l)/.(P + 2)-'-/.(P + ») ""' 
vhere — h^ (p) is the value of the determinant 

/.(p + i). , , Me) . 

/.(p + i), /.(p + 2), ,-., , /.(p) 
/.(p + i). />(>>+2). /.(p+8) , Mf) 

/„(, + !), /.-.(p + 2), /„(p + 3) f,{p + ,-\), /,_,(p) 

/„((.+ !), /.-.(p + 2). /.-.<P + 3) /,((. + "-!). /.W 

SO that h, (p) is a polynomial in p. 

If no two of the roots of the equation /, {p) — differ by whole 
nnmbers, then no denominator in the expressions for the si 
coefficients (j„ vanishes ; the expression £r(«, p) is formally a 
for an integral, but the convergence of the series must 1; 
lished to ensure the significance of the e 



If a group of roots of the equation /„ (p) = differ among one 
another by whole numbers, let them be 

P. p + e^. , p + e, 
where the real part of p i-< the smallest, and that oi p + e is the 
largest, among the real pait** of these roots; equality of roots 
would be indicated by coiiesponding equalities among the positive 
integers 0, e^, ..., e. We then t^ke 

g,-Mf + i)...f.(.P + i)s, 

and thus secure that no one of the coefficients g„ becomes infinite. 
The condition, that the equation shall formally be satisfied, has 
imposed no limitation upon ^j, which accordingly can be regarded 
a'i arbitrary ■ hence tj also can be regarded as arbitrary. 

33 I o ie to Jc 1 v th both sets of cases simultaneously, 
tl e 1 il exp es n s on t ucted in a slightly different manner. 
A ] a n etr c quant ty k s ntroduced and it is made to vary 
wt!n eg s round the oots of /o(p) = 0, each such region round 
a ro t be ng chose o is to c itain no other root. The quantity 
g n tl e hr t Set of c s and the quantity g in the second set, 



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33.] REGULAR INTEGBALS 81 

are arbitrary ; they are made arbitrary functions of a. Quantities 

rji, t^i, ... are determined by the equations 
-»,/.(« + !) + <;,/, (a), 
= ft/.(» + 2) + !7,/,(»+l) + <7./.W, 



the same in form as the earlier equations other than the first : 
these quantities g are functions of a. Moreover, we have 

''■<''>' /.(»-M)/.(»'l1).../.(.+ .;) ''-<°>^ 
in consequence of the assumption as to g^ (a) in the second set of 
cases, and of the regions round the roots of /„(/)) = in which a 
varies, it follows that the quantities gi, g^, ... are each of them 
finite for all variations of a within the regions indicated. We 
thus have an expression 

y=g{x,a)^ 2 g.x-+''-, 
also 

= i^£r.x"+" /<*.« + ") 

= ^„(«)/„(»)^-, 
the coefficient of every power of x except x" vanishing, in conse- 
quence of the law of formation of the quantities g. 

34. We proceed next to consider the convergence of the 
power-series for y, before bringing the equation satisfied by y into 
relation with the original differential equation. We denote by R 
the radius of a circle round the a;-origin within and upon which 
the functions pi, .... pm are holomorphic : so that the circle lies 
within the domain of this origin. Then/(ic, a) and its derivatives 
with regard to x are also holomorphic for values of x within the 
circle and for all values of a considered. As the first of them, say 
/' (x, a), is of degree in a. one less than f{a:, a), it is convenient 
to consider that first derivative : let M (a) be the greatest value 
oi\f' {x, a)| for the values of a; and a, so that, as 



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82 CONVEKGENCE OF THE [34. 

we have* 

and therefore, as k + 1 is a positive integer ^ 1, also 
i/„<»)|«-B-if(«). 
By the definition of the regions of variation of a and the signi- 
ficance of the integer e, it follows that the quantity /o(a + j^ +1) is 
distinct from zero, for all values of w ^ e and for all values of a ; 
hence, as 

from the equations that define the coefficients g, it follows that 

< i/^(.^,^ + i) -| li.!;.l-S-'*(«)+ L?.|-B-+«(" + 1) + ... 

say, where 7^+1 denotes the expression on the right-hand side. 
Evidently 

7.H.,i/.(a + »+ 1)1-7. !/.(«+ >')|J!- = l!;.|«"<« + ») 
« 7, i¥(a + »), 
and therefore 

Let a series of quantities F^ be determined by the equation 

jf(.+») 1 1 /.(«+^) 11 
ii/.(a+rti)rj2|/.(«+»+i)Ir 

for values of I'^e; and let r, = 7,. Then all the quantities F 
thus determined are positive, and we have 

ls'.«l<7.t.<r.„. 

Consider the series 

r,if + r,+,a;-+>+... + r,a!- + ...; 



r.+,-rJ 



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34.] SERIES FOR THE INTEGRAL 83 

its radius of convergence is detennined* as the reciprocfxl of 
Lim -^ , 

Now M{d) is the greatest value of the moduhis of 

- {0 - 1)...(0 - m + 2) p,' - ... - pr^' 
within the circle ]aT| = 7i. As the functions p/, ...,p,„' Eire holo- 
morphic within the circle, there are finite upper limits to the 
values of [pi'l, ..., \pm'\ within the region, say M^, ..., M^; then 

say, where |^| = a. Again 

f,(e)=0(d-i)...(6-m + r)-$(e~r)...(0-m + 2)pAo)-... 

...-P^m, 

ao that, if 

^(»)--„" + ,r(, + l)...(» + m-l) 

+ ,(,7 + l)...(,r + m~2)!p,(0)|+... + lp,(0)|, 
we have 

|/.(fl)|>i9-| -!/.(«)- 9-1 St"- !/.(«)- 9-1, 
and 

the term in 0™ being absent from /, (0) - 0'^, and the term in ff™ 
being absent from i^ (a-). Moreover, as these quantities are 
required for a limit when v tends to infinity, the quantities a- 
and will be large where they occur; thus o-™ is greater than 
(ff), which is a polynomial in a- only of degree m — 1. Hence 

j/. (9)1 S »--*(«). 
Returning now to the expression for r^^^ ^ F^ , let /3 denote 
|al; tlien 

|c< + ,+ l|S>, + l-/3, 
so tiiat 

|a + i.+ l|-S(v+l-(3r. 
Again, 

\a + v+l\tii + X+l3, 
so tliat 

*(|« + i- + l|)«*(» + l + /3), 
and therefore 

j/.(» + « + l)|>(« + l-»--*(..+ l+/3), 

* Chrjstal'e Algebra, vol. ii, p. 150. 



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84 CONVERGENCE OF THE SEEtES [34. 

finally, \a + v\^v + 0, and therefore 

so that 

M{a+ v) ^{" + 0) 

\f,(_ci + v+l)\^iv+l-m'-'-<l>l^ + l + &y 
Now 1^ (rr) is a polynomial in <r of degree m — 1, as also is <j) (o-) ; 
hence, owing to the term (v + 1 — 0)"^ in the denominator on the 
right-hand side, we have 

«(«+» ) _ 
iiri/.(«+»+i)r ■ 

for all values of /9, that is, for all values of a within its regions of 
variation. Again, as /„ (a) is a polynomial in a of degree m, it 
follows that 

.../.(<. + « + !) ■ 
for all the values of a, and therefore 

Using these results, we have 

T . r,„ 1 

and therefore the series 

converges within the circle ja;| = B and for the values of a : conse- 
quently also the series 

7.ic' + 7s+i«'+^-l- ... 
converges for the same ranges of variation for x and a. The 
addition of a limited number of terms that are finite does not 
affect the convergence : and therefore 



converges, for values of x within the circle \a:\ — R, and for values 
of a within its regions of variation. 

Let any region for a be defined by the condition \a-~p\^r. 
Then the series converges absolutely within the a;-circle of radius 
It and the a-circle of radius r. Let R'<R, and r^<r; and let 
K, K denote any finite positive quantities which may be taken 



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34.] INDICIAL EQUATION 85 

small ; then* the series converges uniformly for values of x and a 
siicli that 

\M<«-.. |a-p;<r'-.-. 

Thus the scries converges uniformly in the vicinity of the ai-origin, 
for all values of a in the regions assigned to that parametric 
variable. 

By a theorem due to Weierstrassf, the uniform convergence 
of the series, which is a power-series in x and a function-series in 
a, permits it to be differentiated with regard to a; and the 
derivatives of the series are the derivatives of the function 
represented by the aeries within the a-regions considered. 

Significance of the Indigial Equation. 
35. We now associate the factor x° with the preceding series, 
and then we have 

g (*■, «) = a;- S g^a:' = 2 «/.^+' 

as a series, which converges uniformly within a finite region 
round the ^-origin and can be differentiated with regard to a 
term by term. (It may happen that the origin must be excluded 
from the region of continuity of g (ic, a), as would be the case if 
the real part of a were negative ; the origin must then be excluded 
from the region of continuity of the derivatives with regard to a, 
owing to the presence of terms such as ^„a;°iog3:.) 

The function g{a), a) thus determined has been shewn to 
satisfy the equation 

iij,(=.,<.) = i-/.Wi,.(.). 

As associated with the original differential equation, this result 
requires the consideration of the algebraical equation {hereafter 
called the indicial equation) 

/.((■)=<> 

of degree tn. The preceding analysis indicates that two cases 
have to be discussed, siccording as a root does not, or does, belong 
to a group the members of which differ from one another by 

* Tlie uniform convei^ence with r^ard to a; is known, T. F.,% li,finn. The 
uniform Gonvergenoe with regard to a is eatabliahed bj means of a theorem due to 
Osgood, Ball. Anter. Math. Soc, t. iir (1897), p. 73 ; see the Note, p. 122, at the 
end. of this chapter. 

t Ges. Werhc, t. ii, p. 208; see T. F,. i%S2, 33. 



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86 INTEGRALS ASSOCIATED WITH [35, 

whole numbers (including a difference by zero, so as to take 
account of equal roots). 

Firstly, let p be a simple root ot fo(p) = 0, in the sense that it 
is not equal to any other root and that the difference between p 
and any other root is not a whole number. Then when we take 
a =p, all the coefficients 3i, ^5, ... in 5- (a;, p) are finite ; we have 

that is, 

is an integral of the differential equation : it is associated with 
the simple root p of the equation fo(p)=0, and it is a regular 
integral. 

36. Secondly, let />„, p,, ..., pn constitute a group of roots of 
/o (p) = 0, differing from one another by whole numbers and from 
each of the other roots by quantities that are not whole numbers ; 
and let them be arranged so that the real parts of the successive 
roots decrease : thus the real part of p^ is the greatest and that of 
p„ is the least in the group. In order to secure the finiteness of 
the coefficients^,, ^2, ..., it now is necessaiy to take 

S.(a)-/.(a + l)/.(. + 2) .../.(a + e)j(a), =/(«)<,(.), 
say, where e^p„ — pn, and ^(a) is an arbitrary function of a: and 
now 

Ds, (»,«)- I'S («) n /, (« + ,.)- rj (a) F(„), 
where 

j'W-^n !/.(« + .)). 

Further, there may be equalities among the roots in the group r 
let pu, Pi, pj, pic, ... be the distinct roots taken from the succession 
in the group as they occur, so that po is a root of multiplicity i, 
Pi of multiplicity j - i, pj of multiplicity k —j, and so on. Then in 
^(a), there is a factor {a — p^y through its occurrence in /„(«); 
there is a factor {a — pi)', through the occurrence of (a — piV"^ in 
/o(a), and the occurrence of (a — p,)' in /(,(a + po — pi); there is a 
factor (a — Pj)*, through the occurrence of (a — pj)*~J iny„(a), the 
occurrence of (a — pj)^ in/o{a + pj — pj), and the occurrence of 
(a - pjY in /„ (a + p,- pj). Now 

0<i<j<k<..., 



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36.] A GROUP OP BOOTS 87 

so that, for F(a.), p„ is a root of multiplicity i, that is, 1 <it least : 
pt is a root of multiplicity j, that is, i + 1 at least ; pj is a root of 
multiplicity k, that is, j+ I at least ; and so on. Hence if p, be a 
root in the group as arranged, it is a root of F(a) of multiplicity 
K + 1 at least ; and therefore 

r«>)i .„_ 

L 3»' J. -p. 

for /i=0, 1 K. But 

Be, (.«,«)=»!-<, (.)*■<«), 

and g{3:, a) can be differentiated with regard to a; hence 

= 0, 
for /li = 0, 1 , . . . , K certainly, and for all other integer values of /x. 
less than the multiplicity of p„ as a root of ^(a)=0. Conse- 
quently, the expression 

^ r s^g(^,« )"| 



9p/ 

say, for the same values of fi, provides a set of integrals of the 
equation. 

Moreover, each of the distinct roots in the group thus provides 
a set of integrals; we must therefore enquire how many of the 
integrals out of this aggregate are linearly independent. 

37. We first consider the members of any set ; they are 
furnished by 

for a value p assigned to a, and for a number of values of fi, say 
0,1, ..., K. Kow 

and therefore 

- + ...+(loga;)'Ssr,(a)x'|, 



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88 SUCCESSIVE SETS ASSOCIATED WITH [37. 

where it will be noticed that the coefficient of the highest power 
of logic on the right-hand side is g {x, a). Hence the set is 



y, ^wlogx + w,, 

1/2= w (log xy + 2^1 log x + w^, 

+ icw^-i\ogx + w/,, 

where the coefficients Wp are independent of logarithms. From 
the fact that y^ contains a power of log a: higher than any 
occurring in yo,yi, •■•, ^p-i- it follows (by the lemma in § 27) 
that no linear relation of the form 

c,]/, + c,y, + ... + c^y^ = 

cao subsist among the integrals. 

38. Nest, we consider the sets in turn, associated with the 
values pn, Pi, pj, ... of a, as ai-ranged in decreasing order of real 
parts. The earhest of thorn is given by a = p^ : and it contains 
the i members 






for /i = 0, 1 i-1. Now 

/.(«).(«- p.)' (« - Pi)'-' (■< ~ ft)" ■■■("- Pi)"*-' A . 

*-(.).(«-?.)'(«-?<)* (.t-nf ...(»-p,)"« A,. 

and therefore 

/<" -|lS) " '° - ''•>' <° ^ !'''>' ■ ■ ■ <° ^ l"^'^- 

where A,, A,, A^ are quantities which neither vanish nor become 
infinite for any of the values p„, pi, ..., pi of a Also 

g,(„).g(«}f{a). 

where g{a) is an arbitrary function of a; so that (/o(a) does not 
vanish for a = pn ; a^id therefore the various quantities derived 



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38.] ROOTS IN A GROUP 89 

from ^o(al for 'x = p„, including ^„{a) itself, given by 'l' ^'^^ 
fi = 0, 1, ...,1—1 do not all vanish. Further 

L 3-' J-f. 

which is one of the integrals ; as the quantities 

do not all vanish, this integral belongs to the index p, ; and the 
coefficient of the highest power of log a: is g(x, po)- The iirst set 
thus gives i linearly independent integrals obtained by taking 
fi — 0, 1, ..., i— 1 in the preceding expression. That which arises 
from /I = is 

where all the coefficients are finite : thus it is a constant multiple 
of 

y,*. + aA+i A, (p,) + ^'+^^2 (p„) + . . . , 

an integral that is uniquely determinate. 

Now consider the second set : it is given by a = pi, and it 
contains the members 

for^ = 0. ...,i-l, i,i+l, ...,j-l. The value of5f(a^, a)is 



As regards the first part of this expression, we note that all 
the coefficients gy(a.) for v=0, 1, ..., pa — pi— 1 contain the factor 
(a - pi)* ; and therefore all the derivatives 



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90 FORM OF THE INTEGRALS [38. 

for /t= 0, 1, ..,, i— 1, vanish when a is made equal to pi, while 
they do not necessarily vanish for higher values of fi. 

As regards the second part of the expression for g («, a), we 
wiitc it in the full form 

when a=pi, this becomes 

which accordingly is an integral, and it belongs to the index p„, 
being free from logarithms. But it has been seen that the 
integral, which belongs to the index p^ and is free from logarithms, 
is uniquely determinate, being g (x, p„) ; hence the foregoing 
integral, being the non-vanishing part of g {x, a) when a is p,-, is a 
constant multiple of (/{x, po), say Kg{a;, p„). It might happen 
that K=Q. 

A similar result holds for the derivatives of g(a;, a), for the 
values /J. = l, ..., t — 1. 

Consequently, it follows that the integrals 

for )i — 0, 1, ..., *— 1, can be compounded from the integrals of 
the first set ; they are i in number, but they provide no integrals 
additional to those in the first set ; and thei'efore, without limiting 
the range of their own set, they can be replaced by the i integrals 
of that set. As for the remainder arising from other values of fi, 
they are 

^ r I ^yiM a^+^(|og^) i ^"1^'^-+. . .+(log*y i g^ (p,) wA 
Lv=o 9^-^ ..=0 o/'i* .=0 J 

for /i = i, V + 1, ---,7 — 1- Now 

».(«) = » (•>)<«-?<)'<"- ft)' ■■■ {"-eifA,. 

30 that the quantities 

L 3»- J...; 

for the values s = 0, 1, ..., /t in any one integral, and for the valuea 
fj, = i,i + l, ...,j — l in the different integrals, do not all vanish. 



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38.] IN THE SUCCESSIVE SETS 91 

All these integrals therefore belong to the index pi, and they are 
j — i in number. Moreover, the original set of j integrals, com- 
posed of these j — i and of the replaced i integrals, was a set of 
linearly independent members ; and therefore we now have j — i 
integrals, linearly independent of one another and of the former 
set of i integrals. Thus our second set provides j — i new 
integrals, distinct from those of the first set; and each of them 
belongs to the index p;. The first of them is given by fi — i: 

L.=o 9pi* * .=0 dpi'-' J 

which certainly contains terms not involving log w; ii j — l>i, 
the second of them is 



' y g'^^'g^ (pi) „ 



Ki + l)log.i^?|fc) .- + ...], 



which certainly contains terms multiplying the first power of 
log a;; if _^' — 1 > i + 1, the third of them certainly contains terms 
involving the second power of log x ; and so on. 

The third set among our integrals is connected with the value 
a = pj, and it is given by 

for fi^O, 1, ...,k-l. Now 

The coefficients g, («) contain (a — p^' as factor for all integers v 
which are less than pi—pj', hence the quantities 

vanish for /i = 0, 1, ..., j — 1, and are different from zero only for 
fi=j, j + l, .... k — 1, As in the case of the preceding set, the 
quantities 



\_dai- \ 



.(")- 



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92 FORM OF THE INTEGRALS [38, 

for /i = 0, 1, ..., i — 1, are linearly expressible in terms of the i 
integrals of the first set ; while for fi = i, i + 1, ..., j — 1, they are 
linearly expressible in terms of the j-^i integrals of the second 
set, subject to additive linear combinations of the first set. Thus 
the integrals in the present set which are given by 

r3'9<^.j.)i 

for ^ = 0, 1, ..., ^ — 1, provide no integrals linearly independent 
of the i integrals of the first set and the j —i integrals of the 
second set ; the j integrals in this new aggregate are linearly 
expressible in terms of those in the old. Now the present set of 
integrals, for ;u.= 0,l, ..., j—l, j, j+1, ..., fc— 1, are linearly 
independent of one another; and therefore the integrals for 

/^=j' j+ 1. ■■■- ^'-1 
are linearly independent of one another, of the i integrals of the 
first set, and the j —i integrals of the second set. Thus the third 
set provides k~j new independent integrals, given by the k — j 
highest values of fi. The first of them, determined by f^ = j, is 

which certainly contains terms not involving log a:; if A — 1 >j, 
the second of them, determined by f-=j+ 1, is 



*ft- r S 



8!^(p)^ + 0-+l)logaM. 



which certainly contains terms multiplying the first power of 
loga;; if ft— 1 >^' + 1, the third of them certainly contains terms 
multiplying the second power of log te ; and so on. Moreover, it 
is clear that all these k~j integrals belong to the index pj. 

The law of the successive sets is now clear. The last of thena, 
determined by Oi = pi, contains the integrals 



^^l 



for /i = f, i + 1, ..., n, which are linearly independent of one 
another and of all the integrals of the preceding sets already 
retained. All these integrals, being h + 1 — J, in number, belong 
to the index pi. 



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38.] GENERAL THEOREM 93 

The results thus obtained maj be summarised as follows: 
When the equation fu{p) = hat, a group of roots p„, pi, ..., pn, 
which differ from one anothei hy integers (including zero) and 
differ from all the other toots by quariiities that are not integers ; 
when also the distinct roots aie arranged in decreasing succession of 
real parts, so that p^ is a root of TnuUiplicity i, pi is a root of 
m/idtiplicity j-~i, pjis a root of niultiphoity k —j, and so on, where 
pi' pi' pj' ■■■ ***"* distinct from one another and are arranged in 
decreasing succession of real parts; then, corresponding to that 
group of roots, there exists a group of n + 1 linearly independent 
integrals which are regular in the vicinity of the singularity. This 
group of integrals is composed of a set of i integrals, which are 
given by 

r3;:s_(«^-| 



F^l. 



for fi = 0, 1, ..., i — 1, and belong to the index p„; of a set of j — 
i, which are given by 



for iJ- = i, i + I, --., ? — 1, and belong to the hidea; pi; of a set of 
k—j integrals, which are given by 






for fi =j, i + 1, ..., k—1, and belong to the index pj ; and so on, 
the last set being composed of n + 1 — I integrals, which are 
given by 

I 3«' J„,/ 

for fj. = l, I + 1, ..., n, and belong to the index p;. 

is in substantial agreement with that which 

A different proof is given by iFuchaf: briefly stated, it amounts to the 
establishmont of an integral w,, belonging to the index p,,, to the transforma- 
tion of the equation of order m by the substitution 



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94 INDICIAL EQUATION [38. 

into a linear equation in v of order m- 1, and to the discussion of this new 
equation in a manner similar to that in which the equation of order m is 
discussed. Estiositions of the method devised l>y Fuchs will also be found in 
memoirs by Tannery * and Fabry t. 

39. Ai! the integrals of the differential equation, which has 
the specified form in the vicinity of the singularity, are regular in 
that vicinity ; their particular characteristics are governed by the 
roots of the equation /„ (p) = 0, that is, 

p(p-l)..,(p-m + l)-p(p-l)...(p~m + 2)p,(0)-...-p,(0)=0, 

the differential equation in the vicinity of the singularity being of 
the form 

ax'" ,.„i ^ ^ ' dx"^-^ 

This algebraic equation is of degree m, equal to the order of the 
differential equation; it is calted| the indicial equation of the 
smgularity, and the function f{a;, p), of which fa{p) is the term 
independent of x. is called the indidal function. From the form 
of the integrals which belong to the roots p of the indicial equa- 
tion of a singularity, and those which belong to the roots $ of the 
(§ 13) fundamental equation of the same singularity, it is clear 
that the roots of the two equations can be associated in pairs such 
that 

When the roots of the indicial equation are such that no two of 
them differ by an integer, the roots of the fundamental equation 
are different from one another ; there is a system of m regular 
integrals, and the m members belong to the m different values of 
p. When the indicial equation possesses a group of n roots which 
differ from one another by integers (including zero), the corre- 
sponding root of the fundamental equation is of multiplicity n : 
there is a corresponding group of n regular integrals, the ex- 
pressions of the members of which in the vicinity of the singularity 
may (but do not necessarily) involve integer powers of log x. 
When a root of the indicial equation occurs in multiplicity «, 

* Ann. de 1':Ec. Norm., 2' 86v. t. iv (1875). pp, 113-162. 
+ Thflse, Faculty des Sciences, Paris (188S). 

t Cajley, Coll. Math. Paperi, vol. xil, p. 398. The names adopted by Fuchs 
are determinirende Ftindameittalgleichung, anAdeterminirende Fuiictton, respectively. 



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,39.] LINEAR INDEPENDENCE OF THE INTEGRALS 95 

SO that the corresponding root of the fundamental equation occurs 
in a,t least multiplicity k, there is a set of k associated integrals, 
the. expressions of all but one of which certainly involve integer 
powers of log a;. 

40. Having now obtained the form of integral or integrals 
associated with a root of the indicial equation f^ (p) = 0, we must 
shew that the aggregate of the integrals obtained in association 
with all the roots constitutes a fundamental system. 

First, suppose that the roots of the indicial equation are such 
that no two of them differ by an integer; denoting them by 
^1, pi, ..., pm, and the m integrals associated with these roots 
respectively by wfj, ..., w^, we have 

where Pj (s — a) is a holomorphic function that does not vanish 
when z = tt. No homogeneous linear relation can exist among 
these integrals : for, otherwise, we should have some equation of 
the kind 

Writing 

^, = e2Tip,^ (6- = l, 2, ...,m), 

so that no two of the quantities S,, ..., B^are equal to one another, 
we can, as in § "18, deduce the equation 

c,i?/w, + cA'tVii + . . . + c^Or/w^ = 0, 

for any number of integer values of r, from the above equation, by 
making z describe r times a simple contour round a. Taking the 
latter equation for 7"= 1, ,.., to — 1, the set of m equations can 
exist with values of c,, .,., o™ differing from simultaneous zeros, 
only if 

' , 1 , ..., 1 =0, 



which cannot hold as no two of the quantities are equal. 
Hence we must have e, = = Cs= ... =Cm, and no homogeneous 
linear relation exists : the system of integrals is a fundamental 
system. 



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96 LINEAR INDEPENDENCE OF [40. 

Next, suppose that the roots of the indicial equation caci be 
arranged in seta, such that the members contained in each set 
differ from one another by integers. With each such set of rcots 
a group of integrals is associated, the number of integrals in the 
group being the same as the number of roots in the set. 

It is impossible that any homogeneous linear relation among 
the members of a group can exist: if it could, it would have 
the form 

If Wi, ..,, Wn involve logarithms, then (| 27) the aggregate coeffi- 
cient of the highest power of log (z — a) must vanish ; in the case 
of each integral in which the logarithm occurs, this coefficient 
(§ 25) is itself an integral of the equation, and therefore wo should 
have a relation of the form 

where the quantities w^, ..., w^ belong to different indices, say 
p,., ..., ps, nw two of which are the same; and w^, ..., w„ are free 
from logarithms. Dividing by {z — df>, we should have an equa- 
tion of the form 

hAz-ay'~'''Pr(^ - «) + ■■■ + h>P, (^ - a) = 0, 

where Pr, ■-., Ps ^■'e holomorphic functions oiz — a, not vanishing 
when z = (i. No one of the indices py — p^\& zero ; no two are the 
same : and so the preceding equation can be satisfied identically, 
only \ibr = ■■■ = hs. We therefore remove the corresponding terms 
from 

h{Wi + , . . -1- h,nW„ = 0, 

and proceed as before : we ultimately obtain zero as the only 
: value of each of the coefficients b. 



If Wi, ...,Wm do not involve logarithms, the argument, above 
applied to Wr, -.-, w,, can be repeated: there is no linear relation. 
The initial statement is thus established. 

If the tale of the groups, the members of each of which 
are linearly independent among themselves, is not made up of 
linearly independent integrals, then an equation of the form 
c,w,-|-...-l-c™w„ = 



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40.] THE INTEGRALS 97 

exists. Equating to zero (§ 27) the aggregate coefficient ol' the 
highest power of logs that occurs, we have, as above, a relation of 
the form 

C^Wr + . . . + C,W, + CjilUp + . . . + C5W, + . . . = 0, 
where w^, ..., w, belong to one group, Wj,, ..., w^ belong to another 
group, and so on. Writing 

CrW,4-..- +CsW,= F|, CpWfp+ ... +C5W,, = Ifj, ... 

we have 

Tf, + Tr,+ ...=0. 

Now let 6-i, =6^'"'', be the factor which, after description of a loop 
round a, should be associated with W,; let 6^ be the corresponding 
factor for W^; and so on: the quantities 0i, &^, ... being unequal 
to one another, because TT,, W^, ... belong to different groups. 
Then, as in ^ 18, we deduce the equation 
^,*ri + i9,*F, + ...=0, 
after \ descriptions of the loop; and this would hold for all 
integer values of \. As before, taking a sufficient number of 
these equations for successive values of \, we infer that 

F, = 0, Tr.= o, ... ; 
if these are not evanescent, they would imply relations among the 
members of a group, and so they can be satisfied only if 

c,,= 0=...=c„ 0^ = 0=. ..-c,, .... 
Remove therefore the corresponding terms from the relation 

CiW,+ ... + CmWm=0, 
and proceed as before : we ultimately obtain zero as the only 
possible value of each of the coefficients c. Hence no homogeneous 
linear relation exists; the system is fundamental. 

Some examples illustrating the preceding method of obtaining 
the integrals of a linear equation will now be given. 

Eai. 1. Consider the integrals of the equatioa* 
in the vicinity of the origin. To obtain a regular integral, we take 

* Tho equation is not in the exact form indicated in tlie tast. We have m - 3, 
ji, {0) = 0, and so a factor x haa been removed; alao ive have multiplied hy the factor 



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98 EXAMPLES OF [40. 

substituting, we hare 

provided 

= c,(a2-l)-.o(«+l), 

0=3Caa{a + 2)-2Ci(a + 2)-Co(a-2)3, 

the last holding for n. = 2, 3, . . .. 

The indicial equation is 

.(a-!) = 0, 

giving (simple) roots n = 3, a = 0, so that a f;ictor o + l can be neglected : the 
relations among the coefficients are equivalent to 
(a + 2n-l)ca„^j-C3„=0, 

(a + 2«.)Cs,,5-^„^.= -"-^;n- (-+3i^(f^2« + 2j' 
Firstly, consider tlic root u = 2. We have 

so that the int^ral belonging to the index 2 is 

say the integral is u, where 

Secondly, consider the root o = 0. From the original form of the relations, 

by the first relation ; and the second relation is then identically satisfied, 
leaving c^ arbitrary. Using the reduced form of the relations for the higher 
coefficients, we have 



and therefore the integral belonging to the index is 

On subtracting c^ii from this integral, the reruainder is still an integral, and it 
belongs to the index in the form (say) 

Thus the system of two integrals, regular in the vicinity of x = 0, is 



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40.] THE GENERAL THEORY 99 

This method of dealing with the root a=0 is not quite in accord witii the 
course of the general theory ; it happens to be successful because Cj is left 
arbitrajy. In order to follow the general theory, we note that the coefficient 
of Cj in the original difference-equation cootaina a factor a which vanishes for 
the present root. Hence, taking 



where A ia finite, and so on ; thus 

»-«--(i+.-!i)+'.'-"{>+„-f,+(.+Tf;-s-^j+-}+.-«('..). 

where Ji(i>:, a) ia a holomorphic function of 3: which, by the general theory, 
is finite when a = 0. According to the general theory, this quantity should 
give rise to two integrals, viz. 

Taking account of the value of c^, the first of thorn is zero, thus giving an 
evanescent integral. The second is 

or adding to this itit^ral ^Ca, which is an integral, we have 

C{l-x), 
thus giving 1 - a; as the integral. 

Sx. 3. Discuss iT! a similar manner the regular integrals of the equation 



a the vicinity of the origin : likewise those of the equation 

x(l-x)i'/'-{l + i3!+%3^)«^ + {'3 + Ss:-3^)w^0 
a the same vicinity. 

Ex. 3. Consider the integrals of the equation 

n the vicinity of the origin. Subatituting the espresaion 



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100 bessel's [40. 

provided 

for B = 1 , 2, 3, . . . ; these values give 

where Tis a holomorphic fimotioii otx which is fiaite when a = l. 

The indicial equation has a repeated root a = l; hence two regular 
integrals are 

The former is c^i^:, say the integral is u, where 

the latter is c„a:logx+c„3^, say the integral is e, where 

ii=a: log x + x^. 
Both integrals belong to the index 1 ; and one of them must contain a 
logarithm, since the index is a repeated root of the indicial equation. 

Ei:. 4. Consider Bessel's equation for functions of order zero, viz. 

Substituting 

■u)=o^x''+c,x'^*'-+,..+efie^*''+..., 
we have 



the latter holding for p=i, 2, 3, .... When those relatioi 
value of iw is 



— '•- V (.+2)'^(.+2)-(.+4)' ■■•;■ 

The indicial equation is a*=0, so that o=0 is a repeated root; thus the 
integrals of the eqiuation, both of them helonging to the indes zero, are 

M- [S]„.- 

The firet. of them is 

in effect, J^{x), on making Co=l. The second is 

+«.fi-2i^,(i+«+2rS7iii(i+H5)-...). 



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40.] EQUATIONS 101 

Denoting this by Kf^ wheu Cu = 1 , we have 

where ■^ {p) denotes the value of -j- {log n {:)} when s -p. The two integralu, 
regular in the vicinity of ^ = 0, are J^ and K^. 

Ex. 5. Consider nest Bessel'a equation for functions of order b, via. 
Dw=x^vf' + xv/ ■\-{3^-n?)vi-=i!. 
Substituting an expression 

W = C„:C^-i-Ci3!^*^+ ... +c„x'+"+ ... 

in the equation, we have 

provided 

.,{(a + l}^-«^} = 0, 

for^=l, 9, 3, ... ; we thus have 

r .-0^ 3^ n 

The roota of the indicial equation are 

When n is not an integer, the oorresponding integrals are seen to he 
effectively J„, J.,„. 

When ji IB zero, we have a rei>eated root ; this case has heen discussed in 
the preceding example (Ex. 4), 

When n is an integer different from zero, the two roots belong to a group ; 
and for a — — n, the coeffi.cient of afl^ is formally infinite, so that we have an 
illustration of the general theory in §§ 36—38. We take the roots in order. 

Firstly, let a= +«. : then the integral is 



i (-!)■ 



W n(p)n(«+ii")' 



on taking e^ equal to ---— ^. This is the function usually denoted by J'„ ; 
and it belongs to the index «, 

Secondly, when a= -n, one of the coefficients becomes formally infinite 
through the occurrence of a denominator factor {a + ^nf-it?. Accordingly, 



yGoosle 



bessel's equation [iO. 



and then 



|(„+a«)^_„3;^.ri 



= C{(«+a«)^-«2j^"ri^--^— -,+ ... +{-!)■ 



(a + 2)=-ft= 



n {(« + 2r)=-^=} 



say ; and now 

Two integrals atiae through this root, viz. 

Por the first of them, we have 

SO that it provides no new integral. For the second of them, we have 

raj,,-] 1 2C, •■£Y-''l' "'(''-'^y) ir 

L8»J„— »'n(«-i),;,W n(p) ■ " 

say ; and 

-S?--*''"<'°l="''['-»(in) + -2l(„ + l)(» + 2)l.S--] 

+*^''A 'n(')'n'(wi '■■!'"■'+*'"+'■'-» "■»©" 

say ; so that the integral is M\+ W^, 
In W^, the part represented hy 

is a constant multiple of J„ and therefore can be omitted, owing to the earlier 
retention of ./„. Eejecting this part, and taking 

C=-l2''"'n(j!,-l), 



the integral be(;omes 

_/2Y%-E_fc-pzi)('"'Y' 

U .;. n(j,) w 

+ (f)"J. n4n'(i'+,) '"°^-*'"-H'-'--»@"' 

which differs, only by a constant multiple of ./„, from the espre-swion given by 
Haniel*. 

* Math. AnH,..t. i (186fl|, pp. 469—471, auotod iu my Treatise on Difurentinl 
Equations, p. 167. 



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40.] EXAMPLES 103 

Ex, 6. Discuss in a similar manner the integrals of the equation 
a(l-a-)V + {l~(a + 6 + l)a'}w'-(t6u^=0. 

in the viciniti^ of ;k = 0, and a:=\ : indioating the form for the latter vicinity 

when a+6="l. 

This equation is the differential equation of the hypergeoinetric series 

F{a, b, I, (b). When, in Logendre's equation 

(l-^)^-3,^+y(y+l)».0, 

the indei)endent variable is transformed to a:, where 2=1 — 2j;, it becomes 

a:(l-a-)«;" + (l-2ar)V+i'(F + l)w=0, 
which is the special case of the above given by 6=^+1, a= -p. The 
integrals of L^endre's equation in the vicinity of x=0 and of a;=l, that is, 
in the vicinity of 5=1 and of b=-1, can be deduced from those of the 
hypergeometric equation ; the actual deduction is left aa an esercise. 

Ex. 7. Apply the general theory to obtain the integrals of 
j,^w"' - ^xhe" + Ixw' - 8w = 0, 
which are regular in the vicinity of x=0. 

En. 8. Consider in the same way the equation 

i)()«) = (l+^):cW'-(2+4ic)icW + (4 + 10a!)«w'-(4 + 12;»)«' = 0. 
Substituting for w the expression 

as in the earlier examples, we have 

provided 

C,>(« + a-l)(« + a-S)^ + C„_l(« + a-3)V'i+«-4)=0, 
for!i = l, 2, 3, .... 

The roots of the indicia! equation are 2, repeated, and 1, so that they form 
ft group the members of which differ by integers. Moreover, when = 1, the 
coefficient Cj, which is 

_ (a- 3)^(a-3) 
'-" aW-W ' 
is formally infinite ; for that root, we shall take 
o..C(.-l)'. 
Firstly, for the repeated root n = 2, we have 

W^%X-{\^{a-tfR{x,a% 
where ii is a holomorphic function of x which remains finite when a = a . The 
two integrals are 



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101 EXAMPLES 0¥ [40. 

it is easy to sec that they are constant multiples of 

both of which belong to the itides 2, 

Secondly, for the root a=l, we take a^=C(a—l)^, and theo 

i)(w) = C(a-l)=(<.-2)=:C", 



""". 


-C{a- 


-1)' 


'^- 


-G 




1)'(.^2)'{.- 
.■(- + 1) 


— '. 



— + (a-l)-8(»,.), 

where §(iK, a) is a holomorphic function of .r which remains finite when a = I. 
In connection with this expression, three integrals are derivable, viz. 



'«^-' m.,, [SL.- 



The first of these is 

C2x% 

which is 2C% : it is not a new integral. The second is 

which ia SCm^ -ICu^: it is not a new iotegra,!. The third is 

adding to it liOu^—ZiOui, the new expression is still an int^ral and is a 
constant mulfcijile of %, where 

which manifestly belongs to the iodes 1. 

Ex. 9. Obtain the int^rala of the equations 
(i) (l + a^3W-(Z + i3:^)x^" + (i + lOx^)3^'~{i + 12^-^)v> = 0; 
(i!) (l + 'Lv}3^af'"-(4: + 20x)xhci'" + {U + 72x)x^!i/' 

which are regular in the vicinity of the origin. 
Sx. 10. Consider tiie integrals of 

Ihn=xw"'-i.(aj+bjX+...)w" + {a^ + b^+...)ii/ + {aj + h^ + ...)ir/=0 
in the vicinity of ^ = 0, the constant a, not being an integer. To obtain the 
regular integrals, we substitute 

provided 

for values of n greater than zero, no one of the quantities /j,,/,, ... being of 
degree in n greater than 2. 



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40.] THE GENERAL THEORY 105 

Tlie i-oots of the indicial equation are 

a=0, 1, 2-a,. 

Por a = 2-a[, the difference- equation determines coeificients c„, which 
lead to a series converging for valuea cf |^[ within the common region of 
convei^eni^ of the coefBcients of id", w\ i/>. 

Por «=0 or 1, the difference-equation holds for values of n greater than 
2 or 1 reapcctively ; the only other conditiona are 

o 2 4- -H -0 

fo a= Hi 

^ + =0 

for a=l T) e tl e d fle'en e equat o again detem nes coefBcents wh ch 
in each die leid to a Mr es thit on et^es w th n the -lame regio a. the 
aer e. that helongs to the esp nent 2 o Ea h of the latte tegnl b 
a holo no ph funct on f and tl e efore the three te^iaU of the equat on 
wh h are regula n the v c n ty of j. =0 are — one, a 1 olomo ph t n t on 
of v belong ng to the dex i ^e ond i ke vise i holomorph f net on of % 
belonging to the ndex 1 , and a th rd, belong ng t tl e ndex 2 ^ 

Ex. 11, Discuss the r^ular integrals of the equation in the preceding 
example, when a, is an int^er. 

Ex. 12. Prove that the equation 

has m — i integrali \^hich are holoiiiorphie functinui of % in the vicinity of 
t = 0, when Oj IS not an integer, the various coelhcients a,-i-b^a! + ,., in the 
difterentiil e:iuation being holomorphic m that vicinity, and discuss the 
regulai integrals when •/.^ is -va integer (Poinoare.) 

Ex. 13. Shew that the aeries 

F( r^ = 1+-^r-l- "(" + 1 ) _,.i. 

ya,p,,r,r, ; -^^^^ -^2 ! p (p + 1) ^ {^ + l)r (r + 1) "^■■■ 

satisfies the equation 

.(«,-,„)J+a,; 
and obtain the other integrals, regular in the vicinity of j; = 0. 

Verify that, when a =r, the form of the function F, say (?()), tr,«), satisfies 
the equation of the third order 

and indicate the relation between the two differential equations. 



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106 KEGULAR INTEGRALS [41. 

Regular Integrals, fkee FJiOM Logarithms. 

41. Alike in the general investigation and in the particular 
examples, it has appeared that the regular integrals are sometimes 
affected with logarithms, sometimes free from them. Thus if no 
two of the roots of the indicial equation differ by a whole number, 
each one of the integrals in the vicinity of the singularity is 
certainly free from logarithms ; if a root of the indicial equation is 
a repeated root of multiplicity n, then the first « — 1 powers of 
log a; certainly appear in the group of n integrals which belong 
to that root. When a root of the indicial equation, though not a 
repeated root, belongs to a group the members of which differ 
from one another by whole numbers, the integral belonging to 
the root may or may not involve logarithms r we proceed to find 
the conditions which will secure that every integral belonging to 
that root is free from logarithms. 

Let the group of roots be denoted hy p^^, p^, ..., p^,..., arranged 
in descending order of real parts, so that p, — p„, for k = 0, 1, ... 
fi — 1, is in each case a positive integer: and consider the root 
p^. in order to obtain the conditions under which every integral 
belonging to p^ shall be i'ree from logarithms. In the first place, 
p^ must be a simple root of the indicial equation. Assuming this 
to be the case, we know that the integral belonging to p^ is 

in the notation of § 38. If we further admit the legitimate 
possibility that, to this expression, we may add constant linear 
multiples of the integrals which belong to the earlier roots 
Poi Pi. ---i pi^-i »nd still have an integral belonging to the root 
p^, then, in order to secure that every integral belonging to p^ 
shall be free from logarithms, the . integrals belonging to the 
earlier roots must also be free from logarithms; hence, as 
further conditions, each of the roots p„, p^, ..., p^^, of the indicial 
equation must be simple. These conditions also will be assumed 



The full expression for the integral belonging to p^ is the 
value, when a= p^,. of the expression 



■[J; 



KloS") 2 aTr£f'«' + --+('»g'»)' ^ S.- 



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41.] FREE FBOM LOGARITHMS 107 

in order to be free from logarithms, the quantities 

for (7 = 0, 1, ..., /x- 1, and for all values i' = 0, 1, ... ad inf., must 
vanish : and if these conditions be satisfied, the above expression 
will acquire the desired form. The conditions will be satisfied for 
every value of u, if ^„(a) contains («— p^)" as a factor. But (p. 81) 

g-(«) ^-(°) = rr /„^ 

3„(«) /o(« + l)/o(« + 2).../„(a + l') "'• -*■ 

say; and ^o(a), which (§ 36) is equal to ?(«)/{«), contains 
{a — py,)^ as a factor on account of its occurrence in f{'^)\ 
hence it is sufficient that H,{a) should remain finite (that is, 
not become infinite) when o. = p^, for all values of v. Moreover, 
-ffo(o[)=l. Having regard to the equation by which ^^(a) is 
determined, we obtain the relation 

i/,/.(. + ^)+ir,_./,(. + — 1) + ... 

All the quantities /^{a + v—l), ..., /„(a) are finite for values of a 
that are considered; hence ff„/c(c[ + i') is finite if fl'„(=l), H^, ..., 
/ff_i are finite, and therefore, on the same hypothesis, H^ will he 
finite for all values of v, if it remains finite for those values of the 
positive integer v, which make p^+v& root of the indicial equa- 
tion /5(^)=0. These values are known; in ascending order of 
magnitude, they are 

Consider them in ascending order. We have 



R,. 



When V =/5^_i — p„, a single factor 

/.(« + .) 

in the denominator vanishes when a^p^,,; and it vanishes to the 
first order, because p„_i is a simple root of the indicial equation. 
Hence, in order that jff„ may be finite ibr this value of v when 
a=: pn, it is necessary that 

hy{pi,) = 0, when v = p^^i — p^; 
and it is sufficient that A„(p„) should vanish to the first order. 



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108 REGULAR INTEGRALS [41. 

When V = p^^ - p^, two factors 

in the denominator vanish when a = p^\ and each of them vanishes 
to the first order, because p,_i and p^_j are simple roots of the 
indicial equation. Hence, in order that H, may be finite for this 
value of V when a = p^, it is necessary and sufficient that K{a} 
should vanish to the second order when a = p„i the analytical 
conditions are that 

when V — pi.-2~ Pii ^'Wd a — p^. 

When p = p^_3 — p^, then the three factors 

in the denominator vanish when a — p„; and each of them vanishes 
to the first order, because p^_,, /Jb-s, pn-j are simple roots of the 
indicial equation. Hence, in order that H^ may be finite for this 
value of p when a = p^, it is necessary and sufficient that hy(a) 
should vanish to the third order when a. = p^; the analytical 
conditions are that 

^ ' da da^ 

when v = pn^, — p^ anil a,~p^. 

Proceeding in this way. we obtain the conditions for the 
successive values of :' that need to be considered : the last set 
is that 

~/^ = 0, (<r = 0,l, ...,p-l), 

when V = pa~ Pk ^tid a = p^. 

Such is the aggregate of conditions for a = p^. We have seen 
that, in order to secure the freedom from logarithms of every 
integral belonging to p^,, every preceding integral in the set as 
arranged must similarly be free : and so wo have, in addition, all 
the similar conditions ioT p^_^,p„^^,...,pi, there being no condition 
for the simple root p^. When all these conditions are satisfied, 
every integral belonging to p^ is free from logarithms. 

Manifestly these conditions also secure that every integi-al 
belonging to the roots /?„_,, p^-i, ■■■, pi of the indicial equation 



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41.] FREE FROM LOGARITHMS 109 

is free from logarithms : (one integral, belonging to p„, is always 
unconditionally free from logarithms) : it being assumed that each 
of the roots />o, p], ..., ^^ is a simple root of the indicial equation. 
The conditions thus secure that, when each of the /i + 1 greatest 
roots in the group of roots of the indicial equation is simple, the 
fj,+ l integrals belonging to those roots respectively are free from 
logarithms. 

The preceding investigation is based upon the results obtained by 
Frobcnius, Cretle, t. LXxvl (1873), pp. 224—226. 

A different investigation is given by Fucha, Crelle, t. lsyiii (1868), 
pp. 361—367, 373—378 ; see also Tannery, Ann. de I'tc. Norm., t. iv (187(5), 
pp. 167—170. 

Ex. 1. A simple illustration arises in Ex. 1, § 40, for tlie equation 
3!{2-a?)w" - {x'+Ax.\.'i,){{\ - x)w' -i'w)=0. 
With the notation of the text, wa have 

/.(.)-. (.-2), 
p,-% c-I, Pi-O: 



oonsidcr A„ i 


;a)fora=p, = 0, v = 


Pa- 


-Pi- 


= 2. 


?.(«)^ 


^('■)^o('') 
^/,('' + l)A('< + 2)' 








g.i") 


"a"^^^^"'"'' 









*=(")= ^^^2/1 ("+ ^>-'« <"+ ^^ 

= (4-a)(a + l)(a+2)a. 
The (one) condition m the j re ent t ise w that 

A (qWO 
when 0=0 : which minife^tlj is latished 

Ex. 2. IfthcrtotH f tho indicia! e juati m are different from one another, 
then the integrals whirl beloi g to them certainly possess terms free from 
logarithms. (Fuchs.) 

Ux.Z. Lebpo, pi p le the roota of tie mdiciil equation which form 

a group, the members difieimg by itc^ois and nc two being equal; and 
assume them ranged in descending ordei of icil 1 xrts. Denote po-p, by 
s—l; and form the eqmtion satisfied l> 

V.*,(--'.); 



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110 

then according ( 
tion in W has nc 
integrals of the c 
by logarithms. 



EXISTBKCE OF AN INTEGRAL 

3 the indicia! equation for the singularity a. 
negative roiita or has n^ative roots which 
riginal equation in w are free from li^arithn 



[41. 
= of the equa- 



re aifected 
(Fucha.) 



En. 4. Shew that the integrals of the equations 



(ii) 
(iii) 

where q and 6 arc constants, 



free from logarithms, 
the integrals of the equation 



□ the vicinity of the origin. [They a 



Ki^-h'^f.] 



42. If, instead of requiring (as in § 41) that every integral 
belonging to an exponent p^ shall be free from logarithms, when 
p^ is one of a group of roots of the indicial equation of the type 
indicated in § 36, we consider the possibility that there shall be 
some one integral free from logarithms, belonging to the exponent 
and belonging to no earlier exponent in the group as arranged, no 
such large aggregate of conditions is needed as for the earlier 
requirement. Thus it is no longer necessary to specify that 
po, ..., /3,^i shall be simple roots of the indicial equation; nor is 
it necessary to specify that, even if these roots are simple, the 
integrals associated with them are of the required form. The 
conditions that arise will be particularly associated with a. = p^; 
but they will be affected by modifications arising out of the possible 
multiplicity of />„, ..., p^^i as roots of the indicial equation. 

The detailed results are complicated : a mode of obtaining 
them will be sufficiently indicated by an investigation of the con- 
ditions needed to secure that some integral free from logarithms 
exists belonging to pi and not to pn, with the notation of ^ 36—38, 
Suppose that p^ is a root of the indicial equation of multiplicity i ; 
and let yi, ...,yi denote the set of integrals associated with p^, 
where the expression of y,+], for « = 0, 1, ..., i— 1, is given by 



--P-^l. 



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42] FBEE FROM LOGARITHMS 111 

It' Pi is a root of the indicial equation of multiplicity j — i, only 
the first of the set of associated j — i integrals can be free from 
logarithms: even that this may be the case, conditions will be 
required. Denoting that first integral by W, we have 

Now W certainly belongs to the exponent p;. Its expression, in 
general, involves logarithms ; but there is a possibility of obtaining 
a modification of its expression, so as to free it from logarithms, if 
we associate with W a linear combination of -t/^, ..., yi with con- 
stant coefficients ; and the modified integral will still belong to pi 
but not to /Jo. Accordingly, consider the combination 

where the constant coefficients A are at our disposal ; this gives 



~ xp" 's i 2 !^ t+, , '' ,, (log ^y ^'-Jf; xA . 

(=i„^oj>=(i( ^'pi{t-p)r ^ dp,*" j 

What we require are the conditions that may, if possible, secure 
that no logarithms occur in this expression for U. 

The least aggregate of conditions that will secure this result 
is ; first, 

for all values of v, which secures the disappearance of (log*')'i 
for all values of m and n such that pi + n = p„ + m, as well as 



^_Sp. 



0, 



for p = 0, l,...,p„~pi—l, these conditions securing the dis- 
appearance of (log ic)*~' ; next, 

.■(.-1) ?'.»._ 



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112 CONDITIONS THAT AN INTEGKAL BE [42. 

for all values of m and n such that pi + n = pt, + m, as well as 



for p = 0, 1, ,.., p„ — pi 
appearance of (log ic)'"^ 

ii! dpf 



— 1, these conditions securing the 


dis- 


; next, 








= Ai_,g„ip,) + {i 


-2) A, 


3».. 
"3S 




-.^ 


2 


-2), S-S. 




such that pi + n-- 


-P. + " 


., aa well as 













for all values of m and ) 



for p — 0, 1, . . - , Pa — Pi — 1 y and so on. This aggregate is both 
necessary and sufficient. 

Manifestly any attempt to reduce it to conditions independent 
of the constants A would be exceedingly laborious, even if possible. 
The difficulty arises in even greater measure when we deal with 
the conditions that some integral belonging to p^, where fj, > i, and 
to no earlier index, should exist free from logarithms. 

43. If we assume zero values for all the constants A^, ..., Ai 
in the preceding investigation, the surviving conditions are cer- 
tainly sufficient to secure the result that the integral exists, free 
from logarithms and belonging to its proper exponent: but the 
conditions cannot be declared necessary. 

The aggregate of this set of sufficient conditions is, in the case 
of pi, that the equation 

shall hold for «■= 0, 1, ..., i— 1, and for all values of n. As in 
§ 41, it can be proved that all these conditions will be satisfied if 
the equation 

has a simple root equal to pf. Assuming this to be the case, then 
an integral exists in the form 



■■i.[ij.... 



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43.] FREE FROM LOGARITHMS 113 

which is free from logarithms and belongs to pi (but not to p^) 
as its proper exponent. If pi is a multiple root of the indicial 
equation, the remaining integrals belonging to pi as their proper 
exponent are certainly affected with logarithms. 

Conesponding conditions, that are sufficient (but are more 
than can be declared necessary) to secure the existence of an 
integral, free from logarithms and belonging to an exponent p^ 
(but to no earlier exponent in its group), can similarly be found; 
they are inferred from the investigation in § 41. If the equation 

when n = p^^i — pn, has a simple root equal to p^; if the same 
equation, when n = p^_5 — p^, has a double root equal to p^ ; if the 
same equation, when n = p,^^ — p^, has a triple root equal to p,. ; 
and so on, up to the case of n = pc — p^, when the equation must 
have a root equal to p^ of Kiultipiicity fi. : then an integral exists, 
belonging to pp. as its proper exponent (and not to any of the 
exponents po, p,, ..,, p^^, and free from logarithms. If p^ is a 
multiple root of the indicial equation, the remaining integrals 
belonging to p^, as their proper exponent are certainly affected 
with logarithms. 

On the preceding basis, the identification of the integrals, 
belonging to the group of exponents, with the sub-groups as 
arranged by Hamburger (§§ 23, 24) can be effected. The aggre- 
gate of integrals in the group, which are free from logarithms and 
belong to their proper exponents, not merely indicate the number 
of sub-groups in Hamburger's arrangement but constitute the 
respective first members in the respective sub-groups. The 
general functional forms of the remaining integrals belonging to 
any exponent are (save as to a power of a factor ^Tn) similar to 
those which occur in Jiirgens' form of the integrals in a sub- 
group*. 

44. Ill the practical determination of the integi'ala of specified 
equations, it sometimes"!" '^ convenient to begin with that root 

* In tliie connection, tha followii^ memoirs may be consnlted: Jiii^ena, 
CrelU, t. [JLXX (1875), pp. 150— 16B; SeMesinger, Grelle, t. oxw (1895), pp. 159— 
169, 309—311. 

+ As to this process, see the remarlia by Cayley, Coll. Math. Papers, t. viii. 
pp. 458—162. 



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114 cayley's [44. 

among the group of roots which lias the smallest real part, 
instead of beginning with the root that has the largest real part, 
as in § 36, When the process about to be discussed is effective, 
it has the advantage of indicating at once the number of integrals 
associated with the group which are free from logarithms ; but it 
is not always effective for this purpose, and it does not determine 
the integrals that are affected with logarithms. 

The equations determining the successive coefficients g,, </j, ... 
in the expression 

in the method ot Frobenius are (§ 33) 

= ^«/„(a + K) +?.-,/,(« + «- l)+...+S'».A("). 
for m= 1, 2, .... Let a group of roots of the indicial equation 

/.(«) = 0. 
differing from one another by integers, be denoted by p^, p,, ..., rr, 
where a is the root of the group with the smallest real part ; and 
replace a by «■ in the foregoing typical equation for the y's. Then, 
whenever o- + » is equal to another root of the group, the equation 
in its given form ceases to determine ^„, as a unique finite 
quantity. 

It may happen that the equation is satisfied identically; in 
that case g^ is arbitrary, as well as g^. It may happen that the 
equation appears to determine ^„ as an infinite quantity : in that 
case, we modify ^o as in § 36, and Qn is determinate after the 
modification. 

As often as the former ease arises, we have a new arbitrary 
coefficient; if k be the number of these coefficients loft arbi- 
trary, then K is the number of different integrals, associated with 
the group of roots and free from logarithms. These integrals 
themselves are the quantities multiplying the arbitrary coefficients 
in the expression 

Ex. 1. As an example in which the process, of dealing first with the root 
of a group that has the smallest real part, is efifeotivc as indicating the 



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44.] METHOD 115 

number of integriils free frum Ic^arithms, consider the equation 

The indicial equation is easily found to be 

(p-2)(p-3)S(p-5} = 0, 
th t tt re t ly ill 1 nteRral 1: 1 n n t the exponent 5, free 

fml tlm th m b ml ntegral bei "ing to the exponent 3, 
nd th e w 11 "« ta ly be an t gial, b 1 Dgin t that exponent and 
aff ted th 1 gar th and tl re may be n t gi 1 belonging to the 

ex[K) ent f ee f oia 1 gantl n 

i di 1 t k th alue p=2 ad ub tt t 

= + + + 'f'+ 
th j^ t t th mm diate p pose we d i>t consulor powers 

higher thau fi m v, because p=5 is the root of the mdicial equation with the 
highest real pait The equationa for determming the succes'<ive toefficients 

0=Cj.O+eo.O, 
0-Cj(-2) + c,(2) + to(-l), 
0-Cs.O + C2(-2) + c,(3) + Co(-l); 

from nhiL,h we see that to, <■„ tj lemam aibitrary 111 the other cociB- 
tioita tu^e expressible in terms of them CunsequeotlT, the equation haa 
three mtegiais free fiom lonaiithms helonginn tn 2, \ 'i as their respective 
propel expunents 

(The equation was conotructed so as to haii, 

for a fundamental system ; the system is easily derived by writing 

when the equation for y is 

^{l + j)y"'-aS(7 + 83)y" + j'(294-36«)y'-s(74 + 9fe)y' + (90+120«)^ = 0, 
which can easily be treated by the general method of Frobenius.) 

Ex. 2. As an example in which the process is ineffective, consider the 
Taking, as usual, 



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116 






EXAMPLE 


we have 
provided 




(p+n 




for values 


«=: 


I, 2, .... 





[44. 



If instead, of beginning with the root p=3 fts in tho general theory (§§ 35, 
36), we try p = 2, the equation for the coefficients c gives 

«(«-l)«.-(»-2)>0.-„ 

determiuing c, apparently as infinite. To modify this, we take 

the equation for Cj then becomes 

(p-l)(p-2)c, = (p-3)a0'-2)(?, 

which is satisfied identically, when p = 2. Thus c, rornaina arbitrary ; but 
Co=0, The integral which would be obtained is, in fact, that which belongs 
to p = 3; and the process is ineffective. There happens to be no integral 
belonging to p=2 (and not to p=3) free from logarithms. 

The .Tctual solution ia easily obtained by the general method of Frobeniua. 
"We have 

ir= (7.P ((p - 2) + ^''~_f : + (p - 2)2 (p - 3)= R [z, f,)} , 

where R{z, p) is a holomorphic function of 3 when p is either 2 or 3; and then 

For p = 3, we have the integral 

For p = 2, we have the two integrals 

M)3=ryn =Gz^JrCifi\(igs^ZCz\ 
The integral belonging to the index 3 is 

free from logarithms ; that which belongs to the index 2 is effectively 

2^+2^ log i, 

which is affected with a logarithm, so that the index 2 possesses no proper 
int^raJ free from logarithms. 



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DISCRIMINATION BETWEEN SINGULARITIES 



Discrimination between Real Singularity and Apparent 
Singularity. 

45. The singularity, in the vicinity of which the integrals 
have been considered, is a singularity of coefficients of the equa- 
tion 

da'" z — a di™"^ (z — a)™ 

and the indicoa to which the mtegrals belong are the roots of the 
indicial equation for s = a, which is 



0-0; 



p(p-l)-(p-<n + l} + p(p-l)...{p- 



m+2)P,((j)+... 
.■•+-P.W-0. 
In general, tlie integrals of the equation in the vicinity of a cease 
to bo holomorphic functions of s — a; thus they may involve 
fractional powers or negative powers of z —a, and they may 
involve powers of log (s — a). When this is the case, a is called * 
a real singularity. If, on the contrary, every integral of the 
equation in the vicinity of a is a holomorphic function of s — a, 
then a is called an apparent singularity of the differential equa- 
tion. The conditions that must be satisfied when a singularity of 
the equation is only apparent, so that it is an ordinary point for 
each of the integrals, may be obtained as follows. 

Let Wi, Wa, ..., w^ denote a fundamental system of integrals 
in the vicinity of the singularity a ; and suppose that each member 
of the system is a holomorphic function of 2 — a in that vicinity, 
so that the singularity a is only apparent. Let A denote the 
determinant {§ 10) of this fundamental system, so that 

_ 1 d^^^tVi d''^^Wi dwi ^__ j _ 

dz'"~^ ' dz™~^ > ■•■> ^^ > 

j rf™~'w2 d^~^Ws dwi 

I dz"^'- ' ^2™-^ ' "■' "dz ' 



dw„ 



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118 REAL AND APPARENT [45. 

and let Ar denote the determinant which results from A when the 

column " - j - ^_,' is replaced by —r-^ , (for s=l, ..., m). Then 

as every constituent in A, and A is a holomorphic function of 
^ -a in the vicinity of a, both A^ and A are holomorphic func- 
tions oi z — a in that vicinity; neither of them is infinite there. 
But as in § 31, we have 

m-i' <-> "'>■ 

and some one at least of the quantities P^ (a) is not zero ; hence, 
for that value of r, 

A,(o) 

a(») 

is infinite, and therefore 

A(<.) = 0, 

or the determinant of a fundamental system vanishes at an 
apparent singularity. Moreover, as in § 10, we have 

\dA^_ Pi(^) ^ _ A(tt) dQ(z-- a) 
A dz z — a z— a dz ' 

where ff (s — a) is a holomorphic function oi z — a; whence 

where A\s & constant. Now A is not identically zero near a, for 
the system of integrals is fundamental ; hence A is not zero. We 
have seen that A (a) = 0, and A {z) is a holomorphic function of 
s — a; hence P, (a) m,ust he a negative integer, numerically greater 
than zero. This condition is required, in order to ensure that a 
is a singularity of the equation. 

As each of the integrals is a function, that is holomorphic in 
the vicinity of a, it follows that the respective indices to which 
they belong must be positive integers ; and therefore the roots of 
the indicial equation 

Mp-l)...(p-m+l) + p(p-l)...(p-m+2)P,(tt)+... 

... -|-pP„_i(a) + P„(a) = 

must be positive integers. (When one of these is zero, then 
P^ (tt) vanishes.) Moreover, no two of these roots may be equal ; 



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45.] SINGULARITIES 119 

for otherwise, the expressions for the integrals that belong to the 
repeated root would certainly include logarithms, contrary to the 
current hypothesis. Accordingly, let the roots be p„ p^, ■■■,pm, 
a set of unequal positive integers which we shall assume to be 
ranged in decreasing order of magnitude : they thus form a single 
group the members of which differ from one another by integers. 
The integral belonging to pi involves no logarithm. In order that 
every integral belonging to p^ may involve no logarithm, one 
condition must be satisfied : it is as set out in § 41. In order that 
every integral belonging to ps may involve no logarithm, two 
further conditions must be satisfied ; they are as set out in § 41. 
And so on, for each of the roots in succession until the last; 
in order that every integral belonging to p^^ may involve no 
logarithms, m — 1 further conditions must be satisfied, being the 
conditions set out in § 41. 

The aggregate of these conditions, and the property that the 
roots of the indicial equation are unequal positive integers, give 
the requisite character to the integrals. The condition that P](a) 
is a negative integer makes a a singularity of the differential 
equation. When all the conditions are satisfied, the singularity is 
apparent. 

In all other cases, the singularity is real 



Ex. 1. Consider whether it ia possible that ,r = sliouid be 
apparent singularity of the equation 

where k and X are constants. 

The first condition, that Pi(w) should he equal to a negative ii 
satisfied : in the present instance, it is -4. To discuss the integrals 



only ; 



and substitute : then 

I)v! = Ca{a-i)(a-l)x- 



.,(a + ™-4)(a + «-l)={X(« + «-l) + ^}c„_i, 



The indicia! equation, being (a-4) (a- 1) = 0, has all its roots 
positive integers ; so that another of the conditions is satisfied, 
roots form a group. 



equal to 
The two 



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120 EXAMPLES [45. 

The integral, which belongs to the (greater) root 4 as its index, is a holo- 
morpbic function oi s: ; it is easily proved to be a constant multiple of (say) u, 
where 

"-^t*^ 1.4 ^+ 1.4 ■ 2.5 ^+ 1.4 ■ 2.5 ■ 3.6 ^^-j 
= a^(l +71^4-723^ + 73^ + . ..), 
for brevity. 

As regards the other root given by a=l, we have to assign the conditions 
that the integral whioh belongs to it contains no logarithms. In accordance 
witli the results of § 41, wo seo that there will be a single condition ; 
expr^sing it in the notation there used, we write 

Po=^< Pi = l' '•=Po-Pi=3. f=li 
and we have to find h^{a) for i-=^3, n=p,'=I. Now (§ 38) 
/o(a) = (o-4)(a-l), 

f^ Mffoi") 

/,(a + l)/„(a + 2)/,[« + 3)' 
so that 

A,(fl) = {X(a + 2) + ,c}{\(a + l) + «}{Xa + «}, 
Tho sole condition is that 

and therefore we must have 

K=-\, or -2X, or -3\. 
If K has any one of these values, the origin is only an apparent singularity of 
the equation. 

If K= -X, the independent integi'al belonging to the root 1 is 

If K= -2X, the integral is 
If « = - 3X, the integral is 

In all other cases, the origin is a real singularity of the difierential equation. 

The result, as to the relations between X and /i, can be verified inde- 
pendently. As w and u are solutions of the differential equation, we have 



9M = T 



v/'-mi"=(^ + \\{wv/-mi'), 



and therefore 

where ^ is a constant. Hence 



dw \uj H* (l + yj_^+yiX^ + ys^ + ...f' 



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If every integral is to be holomorphic in the vicinity of the origin, it is easy 
to see that, as M belongs to the indes4, the only condition necessary is that the 

coefficient of - on the right-hand side should be aero. Thus 

wliich, on substitution for yj, y,, y^, and mnltiplication by — 36, givM 

thus verifying the condition obtained by the general method. 

In this example it appears that the integral, which belongs to the smaller 
root of the indioial eq^uation, is, in each of the three possible instances, a 
polynomial in ix. It must not be assumed that such a result always holds 
when a singularity is only apparent ; this is not the case*. 

j&. 2. Prove that the origin is aji apparent singularity for the equation 

where X and /i are constants ; and shew that no integral, holomorphic in the 
vicinity of the origin, can be a polynomial in ;c unless ^ is a positive integer 
multiple of X. 

&. 3. Prove that 3=0 and z= 1 are real singularities for tho equation 
?(l-j)w"+(l-2s)w'-|)i'=0; 
and that 2 = 1, := -1, are real singularities for 

when n is an integer. 

Ex. 4. Shew that s=r» is a real singularity for every equation of the 

where /i {^} denotes a rational function of s. 

£:e. 5. Shew that, if £=<o be an apparent singularity for each integral of 
the equation 

where P and Q are holomorphic functions of £~' for lai^e values of |«|, 
then, if 

!Pf-j = X-l-negative powers of 2, 

-«©-- 

* See some remarks by Gayley, in the memoir quoted on p. 113, note. 



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122 EXAMpr,Es [45. 

X must he a positive integer equal to or greater than 2, and. ft must be a 
positive integer which may be zero. Shew also that, if X = 2, then it must be 

Are these conditions sufficient to secure that each integral of the equation 
is a holomorphic function of 3"' for large values of \z]'i 

Ex. 6. Verify that every integral of the equation 

is holomorphic for lat^e values of \z\. 

Note on § 34, p. So. 

To establish the uniform convergence of a aeries ^.g^^" for values of n, 
Osgood shews {I.e., p. 85) that it is sufficient to h^ve quantities jl/„, indepen- 
dent of a, such that 

provided the series SJf„ converges. 

Take a circle in the a-plane large enough to enclose all the regions round 
the roots of /(p) = given by |n — p|"<j-' — k' ; and let this circle be of radius 
j-j, BO that r^ is a constant independent of a. With the notation of § 34, take 
constants C^, for values of i- ^ t, such that 

f (fi + O 



while G^^T^ = y^. Then, as 

^{v, + v)>M{a-^v\ (-,■, + .)•"-,(,(.■. + .)< |/„(« + . + l)|, 

for all values of v. Now, as in § 34 for the ratios of the r's, we find 

and therefore the series 

converges, li! being leas than It. Accordingly, by taking 

the uniform convei^enee of the series ^ff^^ is established. 



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CHAPTER IV. 

Equations having their Integrals regular in the Vicinity 
OF every Singularity (including Infinity). 

id. We have seen that, if a linear ditfereiitiai equation is to 
have all its integrals regular in the vicinity ot any singularity a, it 
is necessary and sufficient that the equation should be of the form 

cfe™ J— (I rfa™"' {z — af dz™^ " {z — a)™ 
in the vicinity of that singularity, the quantities P,, P^, ..., P,„ 
being holomorphic functions of s - ra in a region round a that 
encloses no other singularity of the equation. We can immediately 
infer the general form of a homogeneous linear differential equa- 
tion -which has all its integrals regular in the vicinity of every 
singularity of the equation, including .2=00. As Fucha was the 
first to give a full discussion* of this class of equations, it is 
sometimes described by his name; the equations are saidf to be 
of Fachsian type or 0/ Fuoksian class. 

Let a,, «£, ..., ttp denote all the singularities of the differential 
equation in the finite part of the ^-plane, and write 

then the conditions are satisfied for each of these singularities by 
the equation 

d"'w_ S Qi, d'"'~'w 

dz™ g=x 1^' dz™"" 

* See his memoir, Crelle, I. lxvi (1806), pp. 139—154. 

+ Care mast be eiercised in order to discrirainate betweea eq'iiitiuas of Fuclisian 
type and Fuehsian equations. The latter arise in eonneetion WLtb automorpMc 
funotions and differential equations having algebraic coefficients ; aee Chap. x. 



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124 EQUATIONS OF [46. 

provided the functions Q, are holomorphic functions of z every- 
where in the finite part of the plane. To secure that the integrals 
possess the assigned characteristics for infinitely large values of z, 
we note that 



+ = ..i!Q, 



where iJ is a polynomial in — and is unity when a = qd , and 
therefore 



= £P"ii" 



©--.©• 



where /ii is of the same polynomial character as B., and is unity 
when 2 = CO . Now suppose that, for very large values of z, the 
determinant A (s) of a fundamental system helongs to the index o-, 
so that 



A(.)=^.r(l), 



where 2" is a regular function of - which does not vanish when 
2 = CO . Then, with the notation of S 31, we have 



A.(»)=^— r.g), 



where T^ is of the same character as T, save that it may possibly 
vanish when z= k: taking account of the latter, we have 

A.W-^~ir.'(i), 

where e is an integer ^0. Thus 



..4='—©. 



where U is a. regular function of - which does not vanish when 
3 = 00 ; and therefore 

Q.-P.r 



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46.] FUCHSIAN TYPE 125 

for very large values of z. But Q^ is a holomorphic function of z 
near z = oo ; this property, imposed on the preceding expression, 
shews* that Q^ is a polynomial in z, of degree not higher than 

(p-i).. 

Moreover, it was proved in the last chapter that all the 
integrals of the equation 

are regular in the vicinity of a = a, when the quantities Pi, ..., Pm 
are holomorphic functions of s in that vicinity. Applying this 
proposition to each of the singularities (including co ) of the 
equation 

with the restriction upon Q,, ..., Q„ as polynomials in s of the 
appropriate degrees, we infer that all its integrals are regular in 
the vicinity of each of the singularities (including x ). 

Combining the results, we have the theorem, due to Fuchs-f: — 
When the m integrals in the fundamental system of a linear 
homogeneous equation of order m. liave a^, a^, ..., ap as the whole of 
their possible singularities in the finite part of the z-plane ; and 
when, all the integrals are regular in the mdnity of each of these 
singularities, as well as for infinitely large values of z; the equation 
is of tfie form 

d^_G^ d-^-'w G^,p^ d"^w G'™,p_i) 

de™' -^ ds'"'^ -yjr^ rfs"'~^ ' 'i|r'" 

where -^ denotes ti (z — a^,and, Oi^if,-!,, for /i = l,2, ...,m,is a 
polynomial in z of degree not higher than fi(p — 1). 

Conversely, all the integrals of this differential equation are 
everywhere regular, whatever be the polynomials G and -i/r of proper 
degree. 

Accordingly, this is the most general form of linear equation of 
order m, which is of Fuchsian type. 

' This result may also be obtained by using the trausformation zx — l and 
applying to the equation, traasformed by the relations in g 5, the proper conditions 
for the immediate vicinity of a; = 0. 

t Crdie, t. i.xvi[186B),p. 146. 



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126 EXAMPLES OF EQUATIONS [46. 

Ex, 1. Legendre's equation is 

(l-32)W-2sju'+m(n+l)ii'=0, 

say 

Its form siatiafiea alt the ueoesaarj conditions ; hence its integrals are regular 
in the vioinity of i=l, z= —1, and are regular also for inflnitely large values 

of 2. 

Similarly, the hypergoomctric equation, which is 

.(l-.).J- + {,-(. + S + l).)«'-a/5».-0, 
has all its integrals regular in the vicinity of ^ = 0, ! = 1, and regular also for 
infinitely large values of z. 

Eessel's equation of order zero is 



__1 ,_£ 
- J «* ^"'■' 

its integrals are r^ular in the vicinity of s=0 ; but, on account of the order 
of the numerator of the coefficient of w in its fractional form, they are not 
regular for infinitely large values of j. 
The same result as the last holds for 

which is JJesael's equation of order «. 

A form of Lanie's equation, which proves useful {see Chap, ix, §§ 148 — 
151), is 

where A and B are constants ; its integrals are regular in the vicinity of any 
point in the finite part of the 3-plane congruent with 3 = 0, and these are all 
the singularities in the finite part of the plane ; but they are not regular 
for infinitely large values of s. 

Ex. 2. The sum of all the exponents associated with all the singularities 
(including oo ) of the equation of Tuchsian type obtained at the end of the 
preceding investigation is the integer |(p- l)ra{m — 1), a result first given by 
Fuchs*. The proof is simple. 

The polynomial ^p-i is of order not higher than j) — 1 : say 
G„-i = A^-^ + .... 
The indicial equation for the singularity a„ is 

ji'^)/]C/9_n...(S-m + 2) + ..., 
' Crdle, t. Livi, p. 145. 



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46.] OF FUCHSIAN TYPE 127 

the unexpresBed terms on the right-hand aide constituting a polynomial in 6 
of order not higher than m — 2. Hence the sum of the indices for the singu- 

and therefore the sum of the indices for all the singularities Oj, a^, ..., (tp in 
the finite part of the plane 

-*'-("-"+ iT?sr 

.Spm(»-1) + J, 
because Oj, a.2, ••■, ^p are tha roots of i^ = 0. 

The indices for co are obtainable by suiistituticg 

the indicial equation for t» is 

{-■irpij>+l)...{p+m-l)={-l)'-'Ap{p+l)...{p+m-2) + ..., 
so that the sum of the indices for co is 

-im(m--l}-A. 
The total suqi of all the indices is therefore 

Hf-l)M(M~-i}. 

Ex. 3. The general eqtiation of Fuchsian type, which haa all ite integrals 
regular in the vicinity of every singularity (including to ), has been obtiiined. 
The limitations upon tlie form of the type are mainly as to degree, so that 
generally the construction of the equations, when definite singularities and 
definite exponents at the singularities are assigned, will leave arbitrary 
elements in the form. The instances when the equations are made com- 
pletely determinate by such an asaignment are easily found. 

Taking the equation as of order m, we have polynomials 

o,_,W, o,_,(.) ff„-.(.) 

which, in their most general form, contain 

p-f-(2p-l) + (3p-2) + ...(mp-m-Hl) 
= ip™(™-i-l)-^m(™-l) 



The assignment of the positions of the singularities merely determines V' : 
it gives no assistance to the determination of the constants in the poly- 
nomials ff. 

Each of the /> singularities in the finite part of the plane requires ™ 
exponents, as does also the point £ = ai ; so that there are ™{p-|-l) constants 
thus provided. But, by the preceding example, their sum is definite : and 
thus the total number of independent constants thus provided is 



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128 EQUATIONS OF FUCHSIAN TVPE [46. 

If therefore the equation is to bo made fully determinate bj the assign- 
ment of these constants, we must have 

and therefore 

ipm(™-l)=J(m-l)(M+3). 

When m=l, p can have any value; that is, any homogeneous hnear 
equation of the iirat order, which has ita integral r^ular in the vicinity of 
each of its singularities and of s = oo , is completely determined by the aaaign- 
ment of singularities and of the exponents for the integral in the vicinity of 
the singularities. 

For such equations of the first order, let a,, ,.,, Op lie th ng 1 t a 
the finite part of the plaoe ; let mj, ..., m^ be the ind to wh h th 
integral beloi^s in their respective vicinities, and let be the d x f 

2=00, so that m+ 2 ^^ = 0. The equation is 



which gives the indes for : = so as equal to - 2 m,. , beiiig it* proper value. 
■yiTien tn>l, then 



ao that, as p is an int^er, m must be 2 and then p=2. Thus the only homo- 
geneous linear equation of order higher than the first, which is of the 
Fuchaian type, and is completely determined by the assignment of the singu- 
larities and of the exponents to which the integrals at the singularities 
belong, is an equation of the second order : it has two singularities in the 
finite part of the plane, and it has a = co for a singularity ; and the sum of the 
sis indices to which the integrals belong, two at each of the singularities, is 
^(2-1)2 (3 - 1), that is, the sum is unity. 

The discussion of the determinate equation of the second order of the 
foregoing type will be resumed later (§§ 47 — 60). 

IiTole. If p =0, so that the equation has no singularities in the finite part 
of the plane, the coefficients are constants if the equation is to be of Fuchsian 
tyX>e. The only singularity of the integrab is at co . 

If p=l, m>l, the number of arbitrary constants is less than the number 
of constants, due from the assignment of the indices at the finite point and 
at 3= CO : the latter cannot then all be assigned at will. 

For values of p greater than 1 and for values of m greater than 1, the 
number of arbitrary constants in a linear differential equation, which are 
left undetermined by the assignment of the singularities and their indices, is 
= ipm{m + l)-im{«,-l)^{m{p + l)-l} 
=i(™-l){™(p-l)-2}, 
which for all the specified values of p and m, other than m = 2 and p = 2 taken 
simultaneously, is greater than zero. 



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Ex. 4. Consider the equation, indicated in the Note to Es, 3, all whose 
integrals arc regular at the only finite singularity, which can be taken at the 
origin, and regular also at infinity ; it is 

'd^" z d^-^ "^ ^ (fe™-2 +■-■+^"'11 

where /j./j, ...,f^ are constants. The assignment of indices for s = determ- 
ines ^, ..., ^, and 80 determines the indices for 3 = co; and similarly the 
aesignment of indices for z = 'a determines those for 2 = 0. In fact, the 
indicial equation for ! = is 

p{p-l)...(p-W + l) = J^p(p-l).,.(p-™ + «-l)/„ 

and the indicial equation for : = (c is 

( \TB{6-\-\) (fl-l- r=2( l)'"-"fl{fl + l) {fi+ K + 1)/" 

t t on e BY deot th fc the ts c n bo a a d 5 a r on fr ta 1 

equat n m the form p+5=0 

As re^rd the ntegrals, t ea y to vo fy n a cordance w th the 

general theory that the ntegral wh h belongs to a s mple r ot of the 
nd al ejuato for =0 a c n t t miltjlo of and that the 

ntegraL wl h bel n t j t i le x)t of tl t e iiiat o re tant 

mult pies of 

for a = 0, 1, ...,«-!. 

Es;. 5. Consider tlie equation 

i>ip = 3{l-3)w" + (l-2s)w'-iw = 0, 
wMch* clearly satisfies the conditions that its integrals should be regular, 
both in the vicinity of its singularities and for large values of :. 



To obtain the i 


utegrals in the vicinity of s=0, we substitute 




)w = eo2"-fc,i!' + i + ... + c„s" + ''-f..., 


and find 






zDiB = a„aH'', 


provided 






{a + «')^c„ = (o+«-^)^c„-i ; 


so that, writing 






f(. + ^){a + |)...(a + m-i)) = 




the value of lu is 





«> = c„s"(l+yi! + y22^ + ...). 

* It ia the differential ec[uatioa of the q^uarter-period in elliptie functions : fo 
detailed discussion of the equation, See Tannery, Ann. Ae Vka, Noi'm. Sup., S^r. S 
t. viii (1879), pp. 169—19*, and Fuohs, Crelle, t. lxsi (X870), pp. 121—136. 



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130 EQUATION OF QUARTER-PERIOD [46. 

Tlie indieial equation is o*=0 : accordingly, the two integrals belong to the 
indes 0, and they are given by 



[a.. 



To particularise the integrals, we take c^^^^tt ; the first of the integrals then 
becomes 

.<„=i.{,.(iy..(H)'--..) 

say : and the second of them becomes Z (s), where 



-| 1 2 *" 2toJ 

say, where 

P™ 1 2^3 4'^"'"^2»i-i 2m' 
And now the two integrals in the vicinity of the origin are 
K(.}, L{f,). 
To obtain the integrals in the vicinity of ^=1, we substitute 

when the equation takes the form 

which is of the same form as in the vicinity of ^ = 0. Accordingly, the 
integrals in the vicinity of 2=1 are given by 

-TW, !■{')■ 
To obtain tbe integrala in the vicinity of 3 = aj, wo substitute 
1 

when the equation takes the form 
The indicial equation for (=0 is 

«("-i)+i-Oi 
and we find the equation for « to bo 



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46.] IN ELLIPTIC rUNCTIONS 131 

of the same form as in the tirat aad the second eases. Accordingly, tha 
integrals of the original equation in the vicinity of a — oo are 

The integrals are thus regular in the vi n ty of tl e t! ve nj, 1 r t cis 
0, 1, to. Of these, the integrals K{z) i(, ) are s gmfl ant n the ioma n 
|e|<I, aay in D^; the integrals K{a!) L{x) ore s gi fic^nt n the domain. 
|a;| = |z-l|<I, say in iJi ; and the integiaU i*A if) ^L{t) aie 3 gn ficant in 
the domain \t\ < I, that is, \z\ > 1, aay in -0„. The ser es A ( ) d verges wi en 
2= I, so that the integrals cease fco be significant for such a value. 

The domains D„ and Di have a common portion, so that values of z esist 
which are defined by 

|a <1, 1J-1|<L 

Within this common portion, the integrals K{z), L{z), K{a:\ L{x) are 
significant : so that, as K {£) aud L (s) make up a fundamental system, we 

K{^)=AK{z)^BL{z), L{:>^)=A'K{z)+B'L{^), 

where A, B, A', B' are constants. The values of the constants are determined 
as follows by Tannery. 

The integrals are compared for real values of a which are positive and 
slightly less than 1, so that, as s then approaches 1, K{e) tends to an infinite 
value. To obtain this infinite value, we note that, as 

by Wallis's theorem, wo have 

and therefore, for I'eal values of s between and I, we have 

The difference of the two quantities, between which the value of K{z) lies, is 
which increases as the real value of s increases and, for a = l, is 
that is, 1 - log 2, Kenofi wc may take 

.,(.)-5iog(i-.). 



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132 EQUATIOK OF QUARTER -PERIOD [46. 

where 

^^>.(3)>j7r-l + log2; 

find the values of z are real, positive, and less than 1. The result shows that 
K(s) is logarithmically infinite for s=l. 

Proceeding similarly with I{z) in the expression for L (s), we have, for real 
values of z between and 1, 

The difference of the two quaatitics, between which the value of iHa) 
s the real value of 3 increases and, for 3=1, is 



and therefore the foregoing difference is less than 

that is, less than (1 - log 2) log 2. Hence we may take 

where, for real positive values of z that are less than 1, 

0<('W<(l-log2)log2. 
The expression can be further modified. Wo have 
S ^^"'<log3 I ^, 

for the values of z considered. The difference between these two series is 
I log 2-^^ 

the real value of s increases and, for 3=1, is 
S ^ (log 2 -£{„). 

8 =-1 '-+. 

"^ 2m + l 2-111 + 2 

1 



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«.] 


IN ELLIPTIC FUNCTIONS 


aod therefore the difference is 




<.!,i^i)<^(l->'>l!^)' 


oil evaluating the ai 


sries. We may therefore take 




J^5^=J_^'^log2-("W 


where 


= -log(l-^)log3-^'(^), 
0<t"(s)<aa-log2). 


Therefore, finally, w 


'0 have 


where 
so that 


i/(.}=-ilog(l-^)log2-.,(4 




0<<i(s)<l-(log3)^; 


and the values of s 


considered are real, positive, and less tl 


In the region co 


mmon to i>„ and jD,, we have 



KIf)-AKif)+BL(i)-. 
and therefore, for real values of z less than (but nearly equal to) 1, that is, for 
real, positive, smaU values of x, 

^(3^)^^*(a)-i^log^-2fllog^log2-4Cf,W+5{.(s)-ilog;i^}loge. 

When s tenda to the value 1, the term log^c log 2 tends to the value : more- 
over, K{!0) then tends to the value ^w ; hence, taking account of the infinite 
terms on the right-hand side, we have 

^-1-45 log 2 = 0. 
Again, when % is real, small, and positive, m ia real, positive, and loss than 
(but nearly equal to) 1 ; hence 

4-W..(»)^ilog(l-»).,(l-.)-ilogz, 



all the terms in which 
when l^l is small. 



{i-^)-iiog2=.iArw+sir(3)ioi 

finite except those i 



J+S/(s>, 
volving lo| 



a holomorphio function of s; thus 
A =1 log 2 ; 



o that A and B a 



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134 EQUATION OF QUARTER-PERIOD [46. 

Similarly, for the other equation 

for values of x and s iu the ooramon region, wo have, for real, positive values 
of z less than I, that is, for rea], positive values of x that are small, 

^Wlog»+/W-^'{,(.)-iIog(l-.))-4ir910E(l-,)log2 + ,,(.)l, 

hence, taking account of the logarithmically infinite terms on both sides, we 

^' + 4S'log2 = -ff, 

Nest, tating the same equation for values of z that are small, real, and 
positive, 80 that x is real, positive, and less than 1, we have 

xir(.)+ff{i-wios.+/(,)).;rWiog «:+/«. 

When a; is nearly unity, 

jrM.,M-iiog(i-»), 

SO that K{3:)logx, for a; nearly equal to 1, is small : and it vanishes when 
a'isl. Also, for those values of ^, 

= -21og3log2-4.,(^); 
whence, equating coefficients, we have 

ijrff=-21og2. 
Thus 

5'= --log 2, ^■=-(log2)=-w. 

Accordingly, when j lies within the portion common to the two domains Df^ 
and Z)j, defined by the relations 

irw.(iiog3)^(.)-ii(.) I 

iM.{"(iog2)>-,}irM-(i log 2) L(.) \ 

where x=l-~e. 

These results shew that, for complex values of j such that [zj^l, both 
X(i) and Z(2) converge. The fiist of them is a known result in the theory 
of elliptic integrals ; writing b = *^ ^-^^ Z(j)=£, AT (^) = Jf', we have 



an equation which is specially useful for small values of h Similarly, for 
Ta]u©a of i nearly equal to unity, we have 



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46.J RIEMANN S P- FUNCTION l3o 

Ex. 6. With the notation of the preceding esample shew that, for values 
of z common to the domains /), and />„ as defined by 

the integrals Z'{a^), i (a'), t^KU), i*Z(;) are connected by the relations 

(Tannery.) 

E^. 7. Denoting the integrals of the equation in Es. S that are associated 

with the values 2 = 0, 1, cc. by ^, Z ; K', L' ; K", L" ; respectively ; denoting 

also the efiect upon a function Uof a, simple cycle round a point ahy \_U^, 

and of simple cycles round a and b in succession by [ U^r, , prove that 

[^li^-fs+f logaW'+^'i'; 
and express [L%, [L"],,, in terms of K', JJ. (Tannery.) 

Ex. 8. Discuss, in the same manner as in Ex, 5, the integrals of the 

(i) ^(l~.)W'-i^=0; 
(ii) 2(l-z)«7" + (l-j)«/ + J«.=0; 
(iii) 2(l-^)W' + w'-iw=0. 

RiEMANN'S P-FUNCl'ION. 

47, It has already been proved {Ex. 3, § 46) that the only 
linear differential equation of any order other than the first, which 
is made completely determinate by the assignment of ita singu- 
larities and of the exponents to which the integrals belong in the 
respective vicinities of those singularities, is an equation of the 
second order which, if ifc have oo for a singularity, has two other 
singularities in the finite part of the plane. If the latter be 
at h,, k, then the transformation 

z—h_ho—b x~a 

z~k k c—a x~b 
gives a, 6, c in the ai-plane as the representatives of h, k; oo in the 
z-plane. The transformation manifestly does not aifect the order 



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136 RIEMANN'S [47. 

of the equation, its sole result being to make a, b, c (but not now 
00 ) singularities ; we shall therefore suppose this transformation 
made. Accordingly, we proceed to consider the properties of the 
function, which thus determines a differential equation ; they 
depend upon the properties initially assigned, which are taken as 
follows. 

In the vicinity of all values of s, except a, b, c (and not 
excepting od when a, b, c are finite), the function is a holooiorphic 
function of the variable. 

In the vicinity of any point (including the three points a, b, c), 
there are two distinct branches of the function ; and all branches 
of the function in the vicinity of any point are such that, between 
any three of them, a linear relation 

AT + A"P" + A'"P"' = 
exists, having constant coefficients A', A", A'". (So far as this 
condition affects the differential equation, it manifestly determines 
the order as equal to two.) 

As exponents are assigned to the three points, let them be a 
and a' for a : /3 and (3' for 6 : y and 7' for c ; these quantities 
being subject (§ 46, Ex. 2) 60 the condition 
a + a' + ^ + S' +y + y=l. 
It further is assumed that a — a.', ^ ~ ^, y — y ai'e not equal to 
integers. The branches distinct from one another in the respective 
vicinities are denoted by P„ and P„. ; Pg and Pp- ; P^ and Pyf. 
From the definition of the exponents to which they belong, the 
functions (s — a)-'P^ and (s - ra)-"'P„' are holomorphic in the 
domain of a and do not vanish when z — a. Similarly for b and c. 

After the earlier assumption, it follows that any branch 
existing in the vicinity of a can be expressed in a form 

where c„ and c^' are constants ; and likewise for branches in the 
vicinity of b and c. The assumption made as to a— a.', /3~0', 
7 — 7' not being integers will, by the results obtained in §§ 35 — 38, 
secure the absence of logarithms from the integrals of the 
differential equation: it manifestly excludes the possibility of 
either of the branches P^ and P^', Pg and P^', Py and Py-, being 
absorbed into the other. 



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47.] p-FUNCTioN 137 

Biemann* denotes the function, which is thus defined, by 



y 



and the function itself is usually called Etemann's P-f unction. It 
is clear that a and a.' are interchangeable without affecting P; 
likewise yS and jS' ; likewise j and 7'. Also, the three vertical 
columns in the symbol can be interchanged among one another 
without affecting P ; six such interchanges are possible. Again, 
if P be multiplied! by (a: - afix-b)-^-' {x -c)', the effect 
is to give a new function, having a singularity at a with expo- 
nents a H- S, a' + 8 : a singularity at b with exponents — B — e, 
iS' — 8 — e ; and a singularity at c with exponents 7 + e, 7' + e. 
Every other point (including 00 ) is of the same character as 
for P. Hence 
/ -a, w r<* ^ '^ 



ia:-hr 



7 



fS /3'- 



-e 7+6 Jr, 
-ey + e J 



the exponents on the right-hand side still satisfying the condition 
that the sum of the exponents shall be equal to unity, 

A homographie transformation of the independent variable 
can always be chosen so as to give any three assigned points 
«', b', c' as the representatives of a, b, c. Accordingly, let such a 
transformation be adopted as will make a and 0, b and <x> , c and 1, 
respectively correspond to one another : it manifestly is 



The indices are transferred to the critical points 0, 00 , 1 ; every 
other point is ordinary for the new function, as every other point 
was for the old. For brevity, the transformed function is denoted by 



' Ges. IVerke, p. 63. 
t The sum o£ the indices in the factor 
a singularity for the new function. 



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where the two-term columns are to be associated with 


order. Also, sir 


ice 




— |5tH. 


it follows that, i 


ixcept as to a constant factor. 




(»-»)'<«-c)- 




(«-!,)•+. '""' " *' "> 


agree ; and thus 


i, as regards general character, we have 


a!''{l~wyF 


1" 1^ 7 A pC +S /5 -«-« T + 
W^'7 / U'+S ,S'-S-6 7' + 



As a — a', /3 — /3', 7 — 7' are the same for the P-function on the 
right-hand side as for the P-funetion on the left, Riemann denotes 
all functions of the type represented by the expression on the 
left by 

P («-«', 0-^', 7-7', ^'). 

In the transformation of the variable, the points a, b, c were 
made to be congruent with 0, » , 1 in the assigned order. A similar 
result would follow if they had been made congruent with 0, 00 , 1 
in any order or, in other words, if 0, 00 , 1 be interchanged among 
themselves by horaographic substitution. As is known, six such 
substitutions a 



or, taking account of the association of the exponents with the 
first arrangement, the table of singularities, exponents, and 
variables for the six cases is 

Oool Owl Oool 

a 13 y x'; 7 /3 a 1-^'; /^ « 7 ^.i 

a' ^' i i ^' d ^' a' y' 



7='^i--'; «7/3 ^3^ ; ^ "y " 1 _"^ ' 

y' a ^' a! 7' /3' /3' 7' a 

so that P-functions of these arguments with properly permuted 
exponents can be associated with one another. 



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48,] i'-FUNCTIONS 139 

48. The significance of the relation 

a + <x' + ^ + 0' + y + y' = l, 
in connection with the function, appears from the following con- 
siderations. When the singularities are taken at 0, x , 1, the axis 
of real variables, stretching from — co to + co , divides the plane 
into two parts in each of which every branch of the function is 
uniform; or, if the singularities be taken at a, b, c, then a circle 
through a, h, c divides the plane in the same way. In either ease, 
taking (say) the positive side of the axis or the inside of the 
circle, the linear relations among the branches of the function 
give 

P, =B,Fe + B,P^.] P. =(7.P^ + 0,Py) 
P„- = B,'P^ + B,Te. j ■ P,< = C'P^ + a^I\. J ' 
Bay 

P„, P„- = iB,,B, $Pp, P^.) = (i^P?, Ps'), 

I Pi', p; I 

p., p.- = ( 6', , C, 5P„ Py) = {cfP-„ Py) ; 

I o;, a; \ 

and with the usual notation of substitutions, lot 
Pg,P^, = (65P„, P,,), 
.Py,Py^ = (ciP.,P.,). 

Consider the effect upon any two branches, say P„ and P,', of 
circuits of the variable round the singularities. 

When it describes positively a circuit round a alone, they 
become e^'''°-P^ and ^'^' P^i respectively, so that, in the above 
notation, 

P„ P,; become ( e^-^, $P„, P.-). 
I 0,«-''| 
When it describes positively a circuit round h alone, then P^ and 
P^i become e'"''^Pp and e^'^'P^^ respectively ; and therefore 

P„ P„. become (fije^'^, J^jPa, P.')- 
I , e^'l 
Similarly, when it desciibes positively a circuit round c alone, 
P., P„- become (cfe'-^y, fc'^^P., P.'). 



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140 RIEMANN'S i'-FUNCTION [48. 

Accordingly, when z describes a simple circuit round a, b, c, tho 
initial branches P^, Pa' are transfonnod into branches 

(oj.-., Vcme-O. jsf,,-, JP..P..), 
, 6''"T'[ I , (f"'f'| I , e=°'"' 

s.y nP.. P..). 

Such a circuit encloses all the singularities of the functions ; and 
therefore* each of the functions returns to its initial value at the 
end of the circuit, so that 

(7)=(1, 0). 

|o. l| 

The determinant of the right-hand side is unity; hence the 
determinant of / is unity, and it is the product of the determinants 
of all the component substitutions. Now as (c) and (c)"' are 
inverse, the product of their determinants is unity ; and likewise, 
the product of the determinants of (b) and {b)~^ is unity. Hence 
we must have 

an equation which is satisfied in virtue of the relation 

a+a' ^ + (i' + y + y'-^l: 
the sum of the exponents could be equal to any integer merely so 
far as the preceding considerations are concerned. 

In the present instance, the property, that a function returns 
to its initial value after the description of a circuit enclosing all 
its singularities, can be used in the form that the effect of a 
positive circuit round c is the same as the effect of a negative 
circuit round a and round b. Applying this to P^, we have 
C,Py^^ + (?,Pye^'" = e-="" (^i^pe-^" + BJ^^.e-"^'^) ; 
and i'rom the expressions for P„, we have 

C\P, + C,Py = B,Pff + B,Pp.. 
As Pp and P^' are linearly independent of one another, it follows 
that e*i™*— eV"* must not be zero, that is, 7 — 7' must not be an 
integer. Similarly for a — a' and 0- &■ 
Ex. Prove, by means of those relations, that 

O,'" S/8in(a'-(-^+y')ir Bj'sin(a'+^'+y),r ' 

£= >-')-*_ A«i°('' + g + y)T _. ga'^iti(n + g' + v)^ 
G4 £,'8m(a' + jy + v)n- iJ2'sin(a'4-(3' + r)7r" 

(Eiemann.) 
* T. F., % yo. 



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49.] determines a differential equation 141 

Differential Equation determined by Eiemann's 
p-function. 

49. As regards the differential equation, associated with these 
P-functions, and determined by the assignment of the three 
singularities a, b, c, and their exponents, we know that it must be 
of the form 
d\v A'z-' + B'z + C 



d^'^{,~a){z-b){z-c)d^^ {z-af{z-bf(z-cr 

which (§ 46) secures that a, b, c, x are points in whose vicinity 
the integrals are regular. Now the singularities are to be merely 
the three points a, b, c, so that oo must be an ordinary point of 
the integral. Taking the most general case, when the value of 
every integral is not necessarily zero for s = x , we have an 
integral 

where Kt, does not vanish. Substituting, we havu 



the unexpressed terms being lower powers of 2; hence 

(2 - A-) K, + A"K, + \B" + 2A" (a + 6 + c)] K„ = 0, 



and so on. Using the result that A" = 0, the equation may 1 
written in the form 

d'-w f A B G\ dw 

U^'^\T-a^~z-b^z-c}~dz 

Forming the indicial equations for the singularities, we have 



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142 DIFFERENTIAL RQUATION DETERMINED [49. 

as the indicial equation for a ; and therefore, as its roots are to be 
a and a', it follows that 

A = 1 —a— a', X= aa' {a—b)(a — c). 
Similarly 

5 = 1-/3-/3', fi = 00'ib-a)(b-c). 

C =1 —y— y, V — 77' (c —a)(c — b). 
Moreover 

A'=A+B + G=% 

on account of the value of the sum of the six exponents; tlio 
condition 

is thus satisfied by B"= 0. All the quantities are thus determined, 
and the equation has the form* 

rf'w n-a-a' l-fi-^'l-j-y'\dw 
dz^ \ z-a z—h z-c ) dz 

[ <.a'{a-h){a-c) &0 (b-a){b -o) 77' (c - a) {c - 6)1 

from the mode of construction we know that the integrals are 
regular in the vicinity of the singularities a, b, c, and are holo- 
morphic for large values of s. This is the differential equation, 
vith (and determined by) the function 
(a b c 



W /3' 1 

The branches of the integral in the vicinity of a are P, , P„-; 
those in the vicinity of b are Pf,, P^i ; and those in the vicinity of 
c are P,, P,-. 

Passing to the ibrm of the function represented by 



U' /3' 7' ' 



where the three singularities are 0, =0 , 1, we deduce the associated 
differential equation from the preceding case by taking 
c( = 0, &=«, c = l; 
• Firat given by Papperitz, Math. Ann., t. xiv (1865), p. 213. 



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id.] BY RIEMAKN'g P-PUKCTION 143 

after a slight reduction, the equation is found to be 

d'w l-ii-a'-(l+i3 + /3');; <i» 

((«■"'' 2(1-2) i 

, W-(aa' + gff'-T7')2 + gi3V „ 

+ 2-(l-2)- 

The branches of the integral in the vicinity of the origin are P,, 
P^', so that ^""'Po, z~°''P^' are holomorphic functions of s, not 
vanishing when 2 = 0; those in the vicinity of a = 1 are Py, Py, 
so that {z — l)'"'Py, {s — l)~*'Py' are holomorphic functions of 
z—l, not vanishing when 2 = 1; and those in the vicinity of 
s — <rj are Pg, Pp., so that z^P^, s^'Pp-are holomorphic functions of 

— , not vanishing when z=<r^ . 

Lastly, passing to the form of the functions included in 
P(t.-«'. /S-/?-, 7-7. »). 
we saw that they arise from the association of arbitrary powers of 
s and 1 — 2 with the above function in the form 

and that they lead to a function 

/. +S, ft-S-,, y+. '. 
-^W + S, ^'-B-,, 7' + .'/- 

Thus we can make any (the same) change on a and a' and, as they 
are interchangeable, we can select either for the determinate 
change; accordingly, we take 



say, as the modified exponents. Similarly, we can make any (the 
same) change on 7 and j : we take 

ry —y = 0, J — y = V — ^ — /i, 

say. Then the new values of the exponents for co are 

/3 + a + 7, -\ say, 
and 



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144 INTEGRALS OF THE EQUATION OF [49. 

on reduction : or the exponents are 

0, 1 - /' , for 2 = , 

\, /t , for s = CO , 

0, v~\-fi, for s = 1 . 
Their sum clearly is unity : moreover, with the preceding hypo- 
theses, the quantities 1-c, /^ —X, v -\ — fi are not integers. 
Specialising the last form of the equation by substituting this set 
of values for a, a', &, ^', 7, 7', we find the equation, after reduction, 
to he 

which is the differential equation of Gauss's hypergeometric series 
with elements X, /t, p. Either from the original form of the 
P-function, or from the resulting form of the equation, the 
quantities X and /j. are interchangeable. 

50. Taking the equation in the more familiar notation 

, dhu , , ^ -, , 1 dw -, „ 

so that tho exponents are 0, 1— 7, for z^O; a, 0, for 3=x; 
0, 7 ~ a — (8, for ^ = 1, we use the preceding method to deduce the 
well-known set of 24 integrals. 

Denoting as usual by F{a:, 13, 7, z) the integral which belongs 
to the exponent zero for the vicinity of z = 0, we have 
«{« + l)M8 + l) 
1.2.t(t+1) 
assignitig to the integral the value um'ty when z= 0. If 

z'(i-^yF{cL',^;y',z) 

be also an integral, then the exponents for each of the critical 
points must be the same as above ; hence 

S, B + l-y =0, 1-7 , for 3 = , 

e, e + j'-a'-^'^O, 7-a-/3, for 2 = 1 , 
a'-S-e, (3'-S-e =«, /3 , for s = oo. 

Apparently there are eight solutions of these equations ; but as a 
and y3 can be interchanged, and likewise a' and 0', there are only 
four independent solutions. These are : — 



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50.] THE HYPERGEOMETEIC SERIES 145 

I. S = 0, e = ; giving a' = a, ^' = /3, 7' = 7 ; and the 
integi'al is 

II. S = l-7, e = 0; giving a' = l + «-7, ^' = 1+^-^, 
7' = 2 — 7 ; and the integral is 
2'-rf (H-a-7, 1+^-7, 2-7, ^); 

III. S = 0, e = 7 — a — /3; giving a' = 7 — n, /3' = 7 — /3, 7' = 7 ; 

and the integral is 

(l_^)v— fljr(^_a, 7-A 7, ^); 

IV. S=l-7, e = 7-a-^; giving a' = 1 - /3, ^' = 1 - a, 

7' = 2 — 7 ; on interchanging the first two elements, 
the integral is 

^'T (1 - z)y---f F(l - «, 1 - A 2 ~ 7, z). 

Next, it has been seen (§ 47) that, in the most general case, 
P-functions can be associated with a given P-function, when the 
argument of the latter is submitted to any of the six homographic 
substitutions which interchange 0, 1, x amoDg one another, 
provided there is the corresponding interchange of exponents. 
Taking the substitution e'z = 1, the new arrangement of exponents 



a, , for / = 0, 






0, 7-a-/?, for s' = l, 






0, 1-7 , for /=co; 
heirce, if 

2"(l-«')--f(«', /3',7'-») 






is an integral, we must have 






S, S + 1 - 7' - 0, ,3 


for /-O, 




e, e+7--ci'-/3'-0, i-a-fi, 


for s'-l, 




a--8-e, /3'-S-. -0, 1-7 , 


fop /-«. 




Again tlrere are four independent solutions ; they are i — 




IX. S-a, «-(); givingf'-a,/3'-l+«- 
and the integral is 


- 7, 7' - 1 + « - 


-13 


z-fU 1 + 11-7, i+«-A 


lY 





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146 


rummer's [50, 


X. 


g = Ae = 0; giving «' = y3,/3' = l+;3-x 7=1 -« + /3; 
and the integral is 




z-^F^^, 1+^-7, l-a+/3, J); 


XI. 


B = ^, e = j-a-0; giving £('=7- a, ^' = 1 - a, 
7' = 1 - a + /3 ; on interchanging a' and ff, the 
integral is 



•(-r"-(' 



XII, S-a, e = 7-a-^; giving a'^^-^, /3'=l-/3, 
7' = 1 + a — ^ ; on interchanging a' and ^', the 
integral is 

s-'(l-^-J~^~^ f(i-^, 7-/3, l+«-A J). 

The remaining four sets, each containing four integrals, and 
belonging to the substitutions 

respectively, can be obtained in a similar manner*. Tliey are ; — 
V. f(o, ft a + ,3-7+1, f); 
VI. (l-f)>^F<«-7+l,/3-7 + l,<. + /3-7 + l, 0; 
VII. fv--«Ji'(7-ii, 7-ft 7-a-3 + l, 0; 
VIII. (1 - f)-» f— « ^(l-a, l-ft 7-«-^ + l, f); 
in which set £; denotes I —n: 

XIII. fj?(«, 7-A «-;3+l, 0; 

XIV. CF{/i, 7-«, /3-« + l. 0; 

XV. (l-i;)"+>?-*'(a-7+l, 1-13, 1-13 + 1, (); 
XVI. (l-t)-"»f'.f'(|3-7+l. 1-ci, /3-1I + 1, a; 
in which set if denotes .j : 

SVII. (l-f)-i?(», 7-ft 7, f); 
XVIII. (l-flBfCft 7-a, 7, 0; 

* The complete set ot expreasions, differently obtained and originally due to 
Knmmer, ate given in my Treatise on Differential EquaUom, (2nd ei.), pp. 192— 
194; the Eoman numbers, used above to specify the cases, are in accord with the 
numbere there used. 



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50.] INTEGRALS 147 

XIX. f-^(i-f)'J?(a-7 + i, :-A 2-,, O; 

XX. i"'{l--i;}'F{/3-i + l. l^c. 2-y. 0; 

in which set f denotes ; and 

s — 1 

XXI. (l-f)"*'(«. <"-7 + I. « + ^-7 + l. E)i 

XXn. (l-f)»F(A^-7 + l, a + /3-7 + l, t); 

xxm. cy'-'0--tyF(i-a, ,,-a, i-a-^ + i. Oi 
XXIV. tr-'(i-O'r(i-i3.j-0. j~„-i3 + i, 0; 

in which set Jf denotes . 

The preceding investigations have been based upon the assumption, 
among others, that no one of the quantities 

is an integer or aero : the determination ot the integrals of the differential 
equation 

when the assumption is not justified, can be effected by the methods of 
§§ 36—38. 

Consider, in particular, the int^rals in the vicinity of s = 0, when l~--y is 
an integer ; there are three cases, according as the int^er is zero, positive, or 
negative. We substitute 

».cy + «, .'+' + . ..+c..'+- + ... 
in the equation ; and we find 

zBto^e{e+y-l)c„/, 
provided 

(^-l +„ + fl)...(n + g)(w-l + g + d)...(g + fl) 
''" (>i + 6) {l + e){n. 1+7+5) (7 + -^) "■ 

(i) Let 1—7 = 0, so that the indiciil equation ls S^ = : then the two 
integrals belong to the mdcs 0, and one of them certainly involves a 
logarithm; and they aie t,nen by 

«-• m.. 

The former, when we take Cq—I, is 

F(a, ft 1, z), 
with the usual notation for the hypoi^eometric function ; as the coefficients 
ai'e required for the other integral, we write 

F{a,0, 1, s) = l+Ki2+«52^+... + -=,.j"+.... 



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148 THE HYPEIIGEOMETRIC [50. 

The second integral, when to it we add 

C(, again being made ec[ual to unity, becomes 

where ij/ (m) denotea ^ {log n (m)). 

(ii) Let I — y be a positive integer, say p, where p>0. The indicial 
equation, being S(S—p)=0, has its root-s equal top, 0. We have 

'^^ (n+e)...ii+6)(n-p+s)...(i-p+e) '■ 

Of the two int^r .Is, that, which belongs to the greater of the two exponents, 
is equal to 

z''F{a-i-p,^+p, i+p,z}, 

when we take <;|,=0. The other integral may or may not involve logarithms. 
If it is not to involve logarithms, then, as in § 41, the numerator of o^ must 
vanish when 8=0, so that 

(p-l+a)...aip-l + 0)...0 
must vanish ; in other words, either a or 3 must be zero or a negative integer 
not less than y. When this condition is satisfied, the integral belonging to 
the index zero is e&ectively a polynomial in z of degree — a or - /3 aa the case 
may be, and it contains a term independent of i. 

When the preceding condition is not satisfled, the integral certainly 
involves logarithms. In accordance with § 36, we take 

Ga=ce, 

so that 



w = (7 S *■■ 



, (^-l + a + g)...(a + fl) (»-l +fl+g)...(g + g ) 
{n + 0)...{l + e) (n-p + e)...{l-p + 0)''' 
There are two integrals givOTi by 



M- [Sh 



The first is easily seen to be a constant multiple of 

z''F{a+p, 0+p, l+p, s), 
thus in effect providing no new integral. The second, after redm 
mating C^l, is 

J-> (»-l+.)....(«-H-|3)...3 
+ „.»!(~-y)(«-l-rt...(l-j.)' 

+(_i,.-i; (— n-.).-.(»-n-w-.g ,. 



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50.] 



=^(^_H.„)+^(«_i+3)->;,(«)-f(»^^). 



(iii) Let 1— 7 be a negative integer, say -q, where 5>0. The indicial 
equation, being ${S-[-q) =0, has its roots equal to 0, - q. We have 
(«-l+„+g)...(„+g)fa-l+g+g)...(3+g) 
(™+6I),..(l+d){«+j+(9)... (1+3+19) ^' 
The greater of the two exponents is 0; the integral which belongs to it, on 
making Cf, = \, becomes 

F{a,»,l-¥q,z). 

Tbfs integral which belongs to the exponent — g may, or may not, involve 
logarithms. If it is not to involve logarithms, then, as before, the numerator 
of tfj must vanish when ^ = - 5, so that 

(.-l)...(a-,;)((i-l)...(e-,) 
must vanish : hence either n or j9 must be a jxisitive integer greater than 
and leas than y( = l + q). When the condition is satisfied, the jnt^ral is 
a polynomial in s~\ beginning with £~', and ending with a~" or 3~P, as the 
case may ba. 

When the preceding condition is not satisfied, the integral certainly 
involves logarithms. As before, in accordance with § 36, we take 
e,~(»+})Jr, 

... „. («~i+.H-»)...(»+») ( «-i+e+<)-0-K ),„i ,,,«H-. 

Two integrals are given by 

The first is easily seen to be a constant multiple of 
/■(n, 3,1+ J, 4 
so that no new integral is thus provided. The second, after reduction, and 
making K= 1, is 

+(-1,.-. J (»-lt -g ).-(°-g)('-l+g-g)-(8-?) ,,— , 

*„=-K«-l+«-?)+^(7i-I+e-?)-v('(»i)-i|'(n-s). 
The integrals are thus obtained in all the cases, when y is an integer. 



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150 EQUATIONS OF THE SECOND ORDER [50. 

Siroilai' treatment can be applied to the integrala of the equ^ition, when 
■y-a— 3 i^ *" integer, positive, zero, or negative, contrary tg the original 
hypothesis as to the exponents for 3=1 ; likewise, when a-,8 is an integer, 
positive, zero, or negative, contrary to the original hypothesis as to the 
exponents for 3=a>. These instances are left as exercises. 

Note. There is a great amount of literature dealing with the 
hypergeometric series, with the linear equation which it satisfies, 
and with the integrals of that equation. The detailed properties 
of the series and all the associated series are of great importance : 
bub as they are developed, they soon pass beyond the range of 
illustrating the general theory of linear differential equations, and 
become the special properties of the particular function. Accord- 
ingly, such properties will not here be discussed: they will be 
found in Klein's lectures JJeher die hypergeometrische Function 
(Gottingen, 1894), where many references to original authorities 
will be found. 



Equations of the Second Order and Fuchsian Type. 

51. No equations of the Fuchsian type, other than those 
already discussed, are made completely determinate merely by 
the assignment of the singularities and their exponents. It is 
expedient to consider one or two instances of equations, which 
shall indicate how far they contain arbitrary elements after singu- 
larities and exponents are assigned. 

Suppose that an equation of the second order has p singulari- 
ties in the finite part of the plane and has co for a singularity ; 
the sum of the exponents which belong to these p + 1 singularities 
is (by Ex. 2, § i6) equal to p — 1. Now let a homogiaphic substi- 
tution be applied to the independent variable and let it be chosen 
so that all the points, congruent to the p+1 singularities, lie in 
the finite part of the plane. Thus « is not a smgulaiity of the 
transformed equation: there are p+1, say re, singulaiities in the 
finite part of the plane: and the adopted transfoimation has not 
affected the exponents, which accordingly aie transferied to the 
respective congruent points. Hence, when an equation of the 
second order and Fuchsian type has n singularities m the finite 
part of the plane and when infinity is not a singularity, the sum 
of the exponents belonging to the n points is equal to n — 2. For 



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51.] AND FUCHSIAN TYPE 151 

such an equation of the second order, let the singularities and 
their exponents be 



then 

Let 

^-.VW = (»-<..)(2-a.).. .<«-«.); 
then, as the equation is of the second order and as all its integrals 
are regular, it is of the form 

where F-^ and F^ are polynomials in z of orders not higher than 
n — \ and 2n — 2 respectively. Also, let 

F _ A, A^ A„ . 

■^ s—a, s — a^ '" s — a„ ' 
and let 

F^ = F^(s) - A"e'^-^ + B"2f^-' + CV-' + .... 

The indicial equation for the point e = a,, is 

'(o-'^ + ^-' + WmT"'' 

and therefore 

a, + S, = l~A„ 



X A,. = n -%(«, + 0,) 

— '2, 

and therefore the polynomial Fi is of the form 

F, = 2s""'' + lower powers of z. 

Again, X is to be an ordinary point of an integral ; hence, talcing 
the most general case, we must have an integral 



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152 EQUATIONS OF [51. 

where K,, is not zero ; for otherwise we should have a special 
limitation that every integral is zero at infinity. Substituting, so 
as to have the equation identically satisfied, and writing 

(so that Sa = 2), we find, as the necessary conditions, 
= K,A", 
= (2 - s,) K, + K,A" + K^ \b" + 2^" 2 a\ , 

<i = {'l> -2s,-\- A")K., + kJ- s, + B" -^lA" i re,) 

+ K„ \a"U I «,' + 2 2 ara)j + 2S" I a, + C'\ , 

and so on. The first gives 

A" = 0; 
then the second gives 



both of these equations leaving Kg and Ki arbitrary. The third 
equation then gives 

and so on, in succession. The remaining coefficients K are 
uniquely determinate; they are linear in Ki and K^, the various 
coefficients involving the singularities and their exponents, as well 
as the coefficients in F^. The equation therefore has z = <x,for 
an ordinary point of its integrals, provided F^is of order not higher 
than 2n — 4. 

The equation can, in this case, be expressed in a different 
form. Let 

^= = 1(0"^-* + ...) 

= Pn--4 + + ^ +,.. + — ^^ , 

s-a, z-as z-a^ 

where P,^^ is a polynomial of order n — i. (Of course, if 2n ~ 4 
is less than n, which is the case when n = 3, there is no such 
polynomial.) As the coefficients in F^ are not subject to any 
further conditions in connection with the nature of 3= co for the 



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51.] FUCHSIAN TYPE 153 

integrals, any values or relations imposed upon \,, X^, .... >.„ and 
the coefficients in Pn-t must be associated with the singularities. 
The equation now is 

^„ /| l-»,-ffA , 1/ |_X, A _„^ 

The indicial equation for a = a, is 

^ (fl - 1) + (1 - a, - ^.) ^ + ^,''^^— ^ = 0, 
and its roots must be ctr, 0r- thus 

and therefore the equation is 

It follows that the only coefficients which remain arbitrary are 
the ji — 3 coefRcienta in the polynomial P,_ (where n ^ 4). When 
the polynomial P„_i is arbitrarily taken, the foregoing is the 
most general form of equation of the second order and of Fuchsian 
type, which has n assigned singularities in the finite part of the 
plane with assigned exponents, and has oo for an ordinary point of 
its integrals. This is the form adopted by Klein*. 

If a new dependent variable y be introduced, defined by the 
relation 

w = y {z — a^Y' {z - a^'"' ... (z-a^)'^, 

then the exponents to which y belongs in the vicinity of a^ are 

the difference of which is the same as for w ; but s = co will 
have become a singularity, unless 

Pi + Pi + ...+p^>0. 
Now 

^1 {(<.. ~ p.) + (0. - p,)) = n ~ 2 - 2 1^ />, ; 

and therefore 

i_ !1 - (», - p,) - (A - p,)i = 2 + 2 l^p,. 

* VorlemngenUberlineareDifferentialgleichungendei-ziueilenOrdnanglGottmgeu, 
1894), p, 7. 



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154 EQUATIONS OF [51. 

Hence, if 3= ic is not to be a siDgularity, the quantities p,, ..., p^ 
cannot all be chosen so that each of the magnitudes 

vanishes. Conversely, if the quantities p^ be chosen so that each 
of these magnitudes vanishes, then z = oo has become a singularity 
of the equation ; having regard to the form of w for large values 
of 2, we see that and 1 are the exponents to which y belongs for 
large values of z ; and the differential equation for y is easily seen 
to be 

where P„_3 is a polynomial of order n~S. 

This equation, however, has n singularities in the finite part 
of the plane, and a specially limited singularity at s = co : we 
proceed, in the next paragraph, to the more genera! case. 

Note. The indicia! equation for a = oo in the case of the 
equation for w is 

0(0 + l)-0j^(l-a.-^.) = O, 
that is. 

The root = gives an integral of the form 
and the root = 1 gives an integral of the form 



^(;-*: 



both of which are holomorphic for large values of |s[, so that all 

integrals are holomorphic functions of - for large values of \z\. 

In this case, oo is not a singularity of the integrals : it can be 
regarded as an apparent singularity of the differential equation, 
and (if we please) we may consider and — 1 as its exponents. 

£x. Shew that the preceding equation can be eshibited in the form 



<!.^^^)-t:.< 



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51.] FUCHSIAN TYPE 155 

where the n constanta c,, ..., o^ satisfy the three relations 

2 c, = 0, S C^r+ 2 c^0r = O, 2 c^/ + 2 2 Orl3^ay = 0, 

and otherwise are arbitrary in the most general case. (Klein.) 

52. Now consider the equation of the second order and of 
Fuchsian type, which has m singularities in the finite part of the 
plane, say Oi, a^, ..., (X«, with exponents a, and /9i, ..., a„ and ^n, 
respectively, and for which m also is a singularity with exponents 
a and /3 : the exponents being subject to the relation 

Let il' denote (z — a^)(s-a^)...(s — an): then the equation is of 
the form 

w"+( I -^"l w' + -^^ w = 0, 

where (? is a polynomial of order not higher than 2n - 2. When 
G is divided by yjr, we have a polynomial of order n-2 and a 
fractional part : and so we may write 

The indicial equatioH for 3 = a^ now is 



4,= l-a, -ft, 
holding for r = 1, 2, . . . , jj. The indicial equation for i 



-l)-0 2 ^, + A„_, = 0, 



-X Ar-1, /(„_ = a^; 



the former being satisfied on account of the relation between the 
exponents. The equation thus is 



w" + 2 - 



.+;..+ s?^^!±>^>l-o, 



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156 A NORMAL FORM OF EQUATIONS [52. 

the coefficients ht,, h^, ..., ^^,-3 being independent of the singu- 
larities and their exponents. 

When a new dependent variable y is defined by the trans- 
formation 

w = (2 -aO"" (^ -«.)"»... (^ - «„)-"y, 

then the exponents ofy for a, are and ^, — a,, say and X^, this 
holding for r=l, 2, ,,., n: and its exponents for x are 

BL+% a,, 0+i a,,, 
= <r, T say : where 

a- + T+ X Xr = ii- 1. 

The function y is, in general character, similar to w: it has the 
same singularities as w, and it is regular in the vicinity of each of 
them but with altered exponents : and it thus satisfies an equa- 
tion of the second order and Fuchsian type, which (after the earlier 
investigation) is 

where i„_3, ..., k„ are independent of the singularities and their 
exponents *. 

This transformation of an equation 



to an equation 

„ , G.-, , ^ 0.-. „ 

where Fa-i, F^_^, G^-i, G„^2 are polynomials of order indicated 
by their subscript index, appears to have been given first by 
Fuchsf, The simplest example of importance occurs for n = 2, 
when the hypergeometric equation is once more obtained. 

53. It is well known that, when y is determined by the 
equation 

y" + Py'+Q = 0, 

* The equation for y can be obtained by the direct substitution of the expreafiion 
for w in tlie eailier differential equation for lu. When reduction takes plaee, there 



t Heffter, Einleitung in die Tkeorie der Unearen DijferentialgUichungen, (1894), 
p. 23i. 



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63.] OF FUCHSIAN TYPE 157 

and a new variable F is introdueed by tbo relation 

the differential equation for ¥ is 

where 

i.e-if-jp.. 

In the case of the preceding equation, the relation between 
y and Y is 

SO that F is a regular integral in the vicinity of all the singular- 
ities and of t» , the exponents being 

i(l-Xr), ^(1+Xr). fo5- s = .a„ ('■ = 1. ■-.™). 

and 

■^ (- 1 + o- - t), ^ (- 1 - o- + t), for s = iO . 

From the form of P and Q, it is easy to see that 

I-i^^ = polynomial of order 2w — 2 

^ L ,.i^-»J 

where P„_2 is a polynomial of order )i — 2, say 

P„ - Bz— + (,_, 2" + . . . + i,. 

In order that ^ (1 — V), ^ (1 + V) niay be the exponents of a^ for 
the equation 

Y" + IT = (1, 
they must be the roots of 

e(e-V) + -,?'--0: 

hence 

JJ,,.1(1-V)^''(«,). 

In order that ^ (— 1 + a- — t), ^ (— 1 — ct + t) may be the exponents 
of <Xi for the same differential equation, they must be the roots of 

0(<f. + l) + C = O: 
hence 

C-ill-(.7^T)-l. 



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158 KLEIN'S NORMAL FORM [53. 

The remaining' constants l^, l^, ---, Ins ^^^ expressible as homo- 
geneous linear functions of &„, i,, .,., k^^i, so that they are inde- 
pendent of the singularities and the exponents : and thus the 
equation is 



-fJ+|[lll-(--)1-" + !.-.^"+...+i. 



r)].o. 



Corollary. For the original equation, x was a singularity 
of the integrals with exponents a- and t. If it were only an 
apparent singularity of the original equation, so that the integrals 
are regular for laj-ge values of \s\, then we have the ease indicated 
in the Note, § 51, so that we can take 

<T, T = 0, -1. 

The modified equation now is 

For this differential equation and its integrals, the exponents to 

which the integrals belong in the vicinity of Or are ^(1 — X,), 

J(l+Xr); but 30 is now a singularity of the integrals, and the 

exponents for a = co are 0, — 1, so that s = i» is a simple zero of 

one of the linearly independent integrals of the modified equation. 

dY 
These forms of the equation, from which tho term in -,- is 

absent, are the normal forms used by Klein. 

The simpleBt example of the class of equations, not made entirely determ- 
inate by the assignment otthe singularities and their exponents, occurs when 
there are three singularities in the finite part of the plane and oo also is a 
singularity. By a homc^raphio transformation of the variahle, two of the 
singularities can he made to occur at and 1, and cc can be left unaltered ; 
let a denote the remaining singularity. Let the exponents he 

0, 1-X(,forz = 0; 0, l-Aifori=l; 0, X for j = u ; 
o-,rfor2=tt>; 
where 

cr-|-T-X„-Xi + X = 0. 

Then the differential equation is 



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53.] EXAMPLES 159 

where q is the (sole) arbitrary constant, left undetermined by the assigned 
properties. The integral of this equation, which is regular in the vioinitj of 
3=0 and belongs to the index 0, is denoted* by 

F{a, q; cr, t, Xj,, X,; s). 

If «•=!, 3=1, the equation degenerates into that of a Gauss's hypergeometrie 
series ; likewise if a = 0, j=0. 

Ex. 1. Verify that, when a = ^, the group of substitutions 

, i_, t ?ri !zi ^ ifci) i!_ 
'■ •■ i- , • .-V .-!• .-1 ' .-i' 

interchanges among themselves the four points 0, ^, 1, co . 

Prove that, when a=~\ and when a =2, there is in each case a corre- 
sponding group of eigbt substitutions interchanging the points 0, 1, a, cd 
among themselves: and that, when a=^{\+i^2) and when a=^{\-i^3), 
there is in each case a corresponding group of twelve substitutions. Construct 
these groups. (Heun.) 

Ex. 2. Prove that there are eight integrals of Heun's equation of tte 

i^{z-\f{z-a)yF{a,q; ,/, r', V. V 5 4 
which are regular in the vicinity of the origin and have the same exponents 
as F{a, q; a; T, X„, X, ; ?). Hence construct a set of 64 integrals for the 
equation when «=J, which correspond to Kummer's set of 24 integrals for 
the hypei^ometric series. 

Indicate the corresponding results when 

a=-\, 3, ^(1 + W3), ja-i^a)- 

Ex. 3. A homogeneouiS linear differential equation of order n is to have n 
singularities ts,, Qj, ,,., a^ in the finite part of the plane and also to have 
00 for a singularity : the integrals are to be regular in the vicinity of each of 
the singularities, and the exponents of 01^ are to be 0, 1, ,.., n — 2, a,, (for 
c=l, ..., «), while the exponents of ro are to be 0, 1, ..., ra— 2, a, so that 

a + J^«, = («-l)^ 

Shew that the differential equation is 



where ij» (s) = (s - Oj) (s - %). , .(j - o,), the coefiicient Eg (b) is a polynomial in z 
of order not greater than », (for 3=1, ..., n), and 

.E,(2)=£(a.-«+l)^-^. 

(Pochh amm er. ) 
' Heun, Math. Ann., t. xxxiii (18891, PP. 161—179, who has developed some of 
the properties of these equations, and has applied them, in another memoir fl.r,., 
pp. 180—196), to Lajnf s functions. 



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160 equations op fuchsian type [54. 

Equations in Mathematical Physics and Equations of 
FucHsiAN Type. 

54. These equations of Fuchsian type include many of the 
differential equations of the second order that occur in mathe- 
matical physics; somebimes such an equation is explicitly of 
Fuchsian type, sometimes it is a limiting form of an equation of 
Fuchsian type. 

One such example has already been indicated, in Legendre's 
differential equation (Ex. 1, § 46). Another rises from a transform- 
ation of Lame's differential equation which (| 148) is of the form 

1 d^ . , , 71 « 
-5J,+^S'W + -B = (l. 

whore A and B ai'O constants*. Writing 

(•(,). c,, 

so that ic is a new independent variable, we liave 

d% / i , J i \diD , , Ax + B „ 



ejdx ' {X 



The singularities of this equation are Bi, e^, e^, oo; the exponents 
to which the integrals belong in the vicinity of Si, e^, e^ are and J, 
in each case ; the exponents, to which they belong for large values 
of a;, are the roots of the equation 

The new equation is of Fuchsian type : and, in this form, it is 
frequently called Lamp's equation. 

An equation, similar to Lamp's equation, but having n singu- 
larities in the finite part of the plane, each of them with and J 
as their exponents, as well as ^ = oo with exponents a and 0, such 
that (§ 52) 

«-^^:=i«~l, 

is sometimes called Lame's generalised equation. By § 52, it is of 
the form 

w" + w' i -J--I- „'^"^° w = 0, 
,^^z a^ Ji(^-a,) 

" This 13 the general iorni; tho -value ~fi(K + l) is assigned (i.e.) to A, in 
order to have those eases of the general form which possesa a uniform integral. 



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5*.] IN MATHEMATICAL PHYSICS 161 

where Gn-i is a polynomial of order n~2, the highest term in 
which is a^s"~^. 

55. The equation of Fuchsian type which, nest after the 
equation determined by Riemann's P-functioii, appears to be of 
moat interest is that for which there are five singularities in the 
finite part of the plane, while ^ = x is an ordinary point. The 
interest is caused by a theorem*, due to EScher, to the effect that 
when the five points are made to coalesce in all possible ways, each 
limiting form of the equation contains, or is equivalent to, one of the 
linear equations of mathematical physics. 

Let the points be a-^, a^, (h, Wj, <h, with indices a^ and ^r, for 
r=l, 2, 3,4, 5; then 

and the equation (p. 153) is 

where -^ = 11 (z — Or), and Pi is a linear polynomial Ax + B. The 
substantially distinct modes of coalescence are :— 

(i), ttj and ttj into one point ; 

(ii), Ma and a^ into one point, a, and a,^ into another ; 

(iii), tta, 0.1, 1X5 into one point ; 

(iv), Oj and a^ into one point, ctj, at, Kj into another; 

(v), at, Ob, ai, a, into one point ; 

(vi), all five into one point ; 
and the various cases will be considered in turn. 



Gase (i). Let the indices for a^, a^, a^ be made 0, ^ for each 
point ; then, as -i/^' (0.4) = 0, 1^' (dj) = in the present case, and 



* Ueber die Eeihenentwiokeluagen i!er Potentialthoorie, GStt. geirSnte Preis- 
sehri/t, (1891), p. 44; and a sepaiate took undpr the same title, p. 193. Sbb also 
Klein, Varlesungtii iiber Uiieare DiJ/ircnhiljl itliuniie?i dee xxaeiten Ordmiiig, (1894), 



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162 bocher's theorem on [55. 

the equation is 

* i F 

Write 

2 — »! = - , (ttr — «4) ^r = !> foi' '■ = 1, 2,3; 
the equation becomes 
d?wdw( i , \ . \ \ , C^ + I> ,.. ^ 

in effect, the preceding ungeneralised Lamp's equation. 

Case (ii). The equation becomes 
^.,^ ^,|l -a.-ft ^ l^K-^8 ' ^ l-.-'-ri 

+(- .-».)(.-';).(.-,.). {^-"^/--''-'''--'4°''- 

itfber coalescence of the points, where 

1-a' -jS' =2-«,-/3,-«,-/3„ 
1 - a" - y9" = 2 - a, - ^, - a, - ,3, , 

and therefore 

fli + /3i + a' + ^' + a" + /3" = 1. 

Writing = (z — a^{z — a.^{z — a,^, we have the coefficient of 

-^ in the form 

z-a, (z- a^) (z - ft.) ' 
where Q,, like P„ is an arbitrary linear polynomial. Thus Q, 
contains two arbitrary coefficients ; these can be determined so 
that 

{z — a-t) {z — at) z — o^ ^ - «j ' 

and then the equation becomes 

.,/■ ^ .„■ | i-''.-ft J. i-«'-g ^ i-»"-ri 



«1 s-o, J-o, a-o, J 



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55.] EQUATIONS OF FUCH8IAN TYPE 163 

Owing to the form of $ and the relation 2 (a + (S) = 1, this is the 
equation of Eiemann's P- function (§ 49). 

When we write ai = l; (1^ = — 1; ai = <xi; 

a„^, = 0, 0; a', ^' = 0,0; a", ^" = ~n,n+l ; 

the equation becomes 

that is, 

{1 - ^)w" - 2zw' + n(n + l)w = 0, 
which is Legendre's equation. 

Case (iii). Let a, , A = 0, i ; a, , A = 0, i ; so that 
1-«,~A + 1 -04-^ + 1 -«>-ft = l. 
After the coalescence of the points, the equation is 

.„"..„.r_i_^.J- 



-a,l (^-o,)(F -»,)(« -,..)■• 



where P) is a linear polynomial, say {^ {z — as) + B}{a, — a^)(as — Oi). 
Now let 



after some easy reduction, the equation becomes 
— , dta f 1 1 



Us — Oj flj — Hj] 



1 



Let 01 = 00, (Xj — tt2 = — 1: the equation is 
d'w , 2ai — 1 (^ A + 



Writing a) = sin'' t, we have 

^-^w{A + B sin= = 0, 



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164 LIMITING FORMS OF AN [55. 

which is known* as the equation of tho elliptic cylinder. This 
equation will be discussed hereafter {^ 138 — 140). 

Case (iv). Let a„ A= 0, i ; '^.03=0, ^; so that, as in the 
1-0,-/3, + ! -0,-^4+1 -a,-^, = l. 
After coalescence of the points, the equation is 

P, 

- (hf' 
Let 

s-«, = i P, = |«(^-«,) + /3}(a,-«,)^ c{a,-a,) = l; 






then the equation becomes 

dHv 1 f^w a + 00) 

or, taking 

we have 

ci% 1 dw 



dhu 1 dw /4a , ^\ 
ay" y dy \y^ J 



which includes Bessel's equation, sometimes called the equation of 
the circular cjKnder. 

Case (v). Let «i, A = ", ^ ; then 

2^(l-«,-^,) = |. 
and the equation, after coalescence of the points, becomes 

«." + a.'(^- + -L~) + ^ ?1 ^^„ = 0. 

Let 

2-„, = i, P, = |»(2-o,) + ,31(a,-o,). 6<»,-o=)-l; 
then the equation is 

dhu dw ^ a + ^x ^ 
da^ aw x — b x -b 

• Heine, Kutjelfunationen, t. i, p. 404. 



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55.] EQUATION OF FUOHSIAN TYPE 

Writing 

the equation becomes 



which is the equation* of the parabolic cylinder. 
Case (vi). The equation is 

w"+ w'+7 ^7W = 0: 

when we take 

the equation becomes 



da? 

This corresponds to no particular equation in mathematical 
physics : it will be recognised as a very special instance of equa- 
tions most simply integrated by definite integralsf. 

Ex. Discuss, in a aimilar manner, the limiting forms which are obtained 
when the singularities of 

(i) the equation determined by Riemann'a P-function, 
(ii) Lamp's equation, expressed as an equation of Fuchsian type, 
are made to coalesce in the various ways that are possible. 

Equations with Integrals that are Polynomials, 

56. There is one simple class of integrals which obey the 
condition of being everywhere regular, so that their differential 
equations are of the Fuchsian type ; it is the class constituted by 
functions which are algebraic. We shall, however, reserve the 
discussion of linear differential equations having algebraic inte- 
grals until the next chapter ; and we proceed to a brief dis 
of a more limited question. 



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166 EQUATIONS WITH [56. 

We have seen that an equation of the second order and of 
Fuchsian type can be transformed to 

By = 1^" + (?„--i/ + G„^y = 0. 
Its integrals are regular in the vicinity of each of n singularities 
and of infinity ; the question arises whether the coefficients in the 
polynomials ©„_, and (?„_3 can be chosen so that one integral of 
the equation at least shall be, not merely fi"ee from logarithms 
or even algebraic, but actually a polynomial in s. This question 
has been answered by Heine* ; the result is that &,^i can be 
taken arbitrarily, and Q.^^ has then a limited number of determ- 
inations. 

If the above equation, in which 

G„_i = CoS"-' + CiS"-^ + . . . + C„_5Z + C^i , 

(?„_3 = k^-^ + h^"^ + . . . + k^^ + ^„_2, 

is satisfied by a polynomial of order ni, say by 

y ^ g^"^ + gi^^'"- + ■■• +9^-1^ + 9,^, 
then 

80 that there are m + n — l relations among constants. The form 
of these relations shews that gi, g^, ..., gm ^re multiples of ff,,'- to 
express these multiples, m of the relations are required, and when 
the values obtained are substituted in the remainder, we have 
n — 1 relations left, involving the constants c and k. Assuming 
the points ai, a^, ..., a„ arbitrarily taken, and the coefficients 
Co, Ci, .,., c„_] arbitrarily assigned, we shall have these n—1 rela- 
tions independent of one another, and therefore sufficient for the 
determination of the K — 1 constants /c^, ^i, ...,^»_a. 

The first of these relations is 

m(m-l) + o,m + k, = 0, 
so that ks is uniquely determinate. Denoting by 

[kuh. ...,K]r 
the generic expression of a function of ki, k^, ..., k^, which is 
polynomial in those quantities, and the terms of highest weight in 

* Heine, Kugel/unctioneii, t. i, p. 473. 



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56.] POLYNOMIAL I^fTEGRALS 167 

which are of weight r, when weights 1, 2, . . . , u — 2 are assigned 
to ^1,^2, ..., fca-s, we have, from the m relations next after the 
first, 

for )■= 1, 2, .... m. When these are substituted in the remaining 
w — 2 relations, we have 

for s=l, 2, ..., « — 2. These determine the m — 2 constants 
fci, ^2, ...,kn-2', the number of determinations may be obtained 
as follows. Writing 

the equations become n - 2 equations to determine w — 2 quantities 
Xi, (c^, ,.., «B-a. In these quantities, the equations are of degrees 

m + l,m + 2, ...,m+m-2, 
respectively; and therefore the number of sets of values for 
iCi, x^, ..., Xn-2 is 

(m+l)(m + 2)...(m + «-2). 

But the same value of k^ is given by two values of x^, inde- 
pendently of the other constants k; so that the sets of values 
of «i. a's, ■•-, iTn-amust range themselves in twos on this account. 
Similarly, the same value of k^ is given by three values of iCj, 
independently of the other constants k ; hence the arranged sets 
of values must further range themselves in threes, on account of 
kj. And so on, up to k^^. Hence, finally, the number of sets of 
values of ^1, ..., kn--^ is 

(m + l)(m + 2)...(m + «-2) 
2.3...JI-2 

^ (ffi + H-2)l 

~ m 1 (n - 2) I ' 
which therefore is the number of different quantities Gn~i per- 
mitting the equation 

to possess* a polynomial integral of degree in. 

* In Gonnection with these equationa, a memoir by Humbert, Jourw. de I'Ecole 
Polytechaique, t. jliix (1880), pp. 207—220, may be consulted. 



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168 POLYNOMIAL INTEGRALS [56. 

This result is of importance, as being related to those special 
forms of Lame's differential equation which possess an integral 
expressible as a polynomial in an appropriate variable. This 
polynomial can be taken as one of the regular integrals belonging 
to each of the singularities ; the other regular integral belonging 
to any singularity is, in general, a transcendental function and, in 
general, it involves a logarithm in its expression. 

Es!. I. Shew that a, linear equation of the third order, having all ita 
integrals regular, caii, by appropriate transformation of its dependent variable, 
be changed to the form 

where 

>/.=(^-a,)(e-<i,)...(E-<i.), 

Ill, itj, ..., On being all the singularities in the finite part of the i-plane, and 
■where P, Q, B are polynomial functions in z of degrees ii-l, 2ii-2, 2ji— 3 
leepectively. 

Shew that, if P and § be arbitrarily chosen, R can be determined so that 
one integral of the equation is a polynomial in i ; and prove that the number 
of distinct values of B is 

(m + 2»-3) ! 
m\(,2n-sy.' 
where m is the degree of the polynomial integral. 

Es. 2. Determine the conditions to be satisfied if 

has two distinct polynomials as integrals, so that every integral is a poly- 
nomial. 

Ex. 3. Determine how far the constants in the equation 

may be assumed arbitrarily if the equation is to possess two polynomial 
integrals. 

Sc. i. Prove that the equation 

/wg+l/'w|-t(.(.+ i)«*l»-o 

■where »i is an integer, f(x)~x^+a3fi+ba:+c, and a, b, c are constants, admits 
of two integrals whose product is a polynomial in x. 

Ex. 5, Shew that the only cases, in which the differential equation of the 

.(i--)2+(v-(.+f.+i)«)*-.»=o 



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EXAMPLES 169 

s whose product is a polynomial in ie of degree n, are as 
follows. If n is an evea integer, then either a=-^n; or j3=~Jn; or 
a+/3=-m, and y = ^, or -^, or -|, ..., or -re+i. If ra is aJi odd integer, 
then either a=-\n and y=i, or -|, or -f, ..,, or -|w+I, or ft or 3~1, 
..., or »^\{n~\); or fi=-\r<, and 7=^, or -i, or -f, ..., or -i»i+l, 
ora,ora-l, ...,or<.-i(™-l); or a4-j3= -«, and y=i, or -i,or -§,..., 
or -)i+^: (Markoff.) 

Sr. 6. Sliew tliat, if the square root of a polynomial of degree m can be 
an integral of tte equation 









(V- 2 Xrf^,)^'""^+''i^" 


^+. 


.+««-2 


n^(;^-0 







whore the exponents X and p. are subject to the usual relation, one of the 
exponents \„ ^„ say X„ must be half of a non-iiegative integer, this holding 
for each value of s ; also ^ra — 2X, must be a non-negative integer; and one 
exponent of the singularity at iofinity must be equal to - ^»!. 

If these conditions are satisfied, how many such equations exist? 



n Vietk.) 



Ex, 7. If the differential equation 



"■" -"•''"• 11 (»-.,) 

where ^(^) is a polynomial, the constants a are real and positive, and tlie 
Constanta e are real and distinct from one another, be satisfied by a poly- 
nomial ^ {x), then all the roots of (.c) are real, and no toot is leas than the 
least or greater than the greatest of th.e quantities e. (Stieltjes ; BSoher.) 



Equations with Rational Integrals, 

■37. The investigation in § 56 suggests another question: 
what are those linear equations, all the integrals of which are 
rational naeromorphic functions of 3? 

Let a,, ..., Oto he the singularities in the finite part of the 
plane; let a^t, «^, ■•-, a«r be the roots of the indicial equation for 
Of-, and let /3i, ..., 8n he the roots of the indicial equation for 
z — x>. If every integral is to be a rational function of z, all the 
roots a,r, Owi ■-■, a„r must he integers; as no integral is to involve 
a logarithm, no two of them may be equal. Let the arrangement 



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170 EQUATIONS WITH [57. 

of these roots be in decreasing order of the integers. The integral 
belonging to the index a,r involves no logarithms ; in order that 
the integrals belonging to the indices a„, a^r, ■■■, a»r respectively 
may involve no logarithms, 

H-2+...+(n-l), 
that is, \n{'ii—l), conditions in all must be satisfied, these 
conditions being as set out in § 41. Corresponding conditions 
hold for each of the singularities, and also for «= « ; so that there 

i»(«-l)(m + l) 
conditions of relation among the constants of the equation, in 
addition to the necessity that the indicial equation of each singu- 
larity shall have unequal integers for its roots. 

These conditions are certainly necessary ; they are also suffi- 
cient to secure that any integral of the equation is a rational 
function of z. For considering the vicinity of a,, each integral in 
that vicinity is of the form 

where «„,. is the least of the roots of the indicial equation, and 
Pm(3 — t,) is holomorphic in the vicinity of a,., for m= 1, ..., «; 
when m = «, P{z —Of) does not vanish, and for all other values 
of m it does vanish. If then o„r be zero or positive, the point 
s = », is an ordinary point for every integral in the vicinity of «, ; 
if Ojir be negative, then a,, is a pole of some integral, and it may be 
a pole of several or of all. 

As this holds in the vicinity of each of the singulaiities and of 
s = 00 , it follows that, in the vicinity of every singularity of the 
equation, including z — ca, every integral is uniform and has that 
singularity either for an ordinary point or a pole ; moreover, every 
integral is synectic in the vicinity of every other point : hence* 
the integral is a rational function, which is a polynomial if oo be 
the only pole. Thus the conditions are necessary and sufficient. 

It has been seen that the indicial equation for each singularity 
of the differential equation must have unequal integers for its 
roots. When these are assigned arbiti-arily, subject to the one 
relation (Ex, 2, § 46) which they are bound to satisfy, they amount 



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57.] RATIONAL INTEQEtAXS 171 

to (m + 1) K - 1 conditions ; so that the total number of necessary 
conditions is 

^n{n-l){m + l) + (m + l)n-l 

= in(n + l)(m + l)-l. 
If such equations are being constructed, they are necessarily of 
the form 



where -^ = {z -a^ ...{z~ a»), and 0^ is a polynomial of order not 
greater than r{m— 1), for r= 1, ...,«. Henue the total number 
of disposable constants is 

m, from the positions of the singularities, 

+ S \r{m-\)+ 1}, from the constant coefficients in Gi, ..., G„, 

that is, 

i»(n + l)(m-l) + » + ». 

constants in all ; and therefore, in order that the equations may 
exist, we must have 

\n(n + \){m-\) + n + m>\nin + X)(m + l)-l, 

so that 

m^K^-1. 

In obtaining this result, an arbitrary assignment of unequal 
integers as roots of the indicial equations has been made : and 
it has been assumed that these conditions are independent of the 
necessary conditions attaching to the coefficients, in order that 
the integrals of the equation may be free from iogarithms. It 
may, however, happen that a particular assignment does not leave 
all these conditions independent of one another, so that we might 
have 

i»(» + l)(».-l)+« + m.in(»+l)(m+l)-l-\, 

and therefore 

m = 7i'-l-\, 

and still have the equation determinate. An instance is furnished 
by the equation 



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172 EQUATIONS WITH [57. 

which, although it has only one singularity in the finite part of 
the plane, so that m = 1, i! = 2, has an integral Ax"^ + Bx. For the 
most general case, however, we have 



JSr. 1. Investigate all the oasea in which the differential equation of the 
hypergQometric series haa every integral a I'ational function of the independ- 
ent variable. 

fiiir. 2. When the equation is of the second order, and all the assignments 
of integer roots are quite general, the smallest value of m is 3. Let the 
singularities be «[, ..., %,, with exponents aj, S,; a^, |9ji ...; a^, ft„; and 
let the exponent* for ^= a> be a, S- Choosing in each case the smaller of the 
two indices Or and (9r, let it he o,., for r = l, ..,,nt; then writing 

\r = »,-ar, a+ S n^ = <r, 3+ S o^=r, 
we have (§ 52) 

cr + r+ S \^m-\, 

which is the necessary relation among the exponents. Writing 
so that y also is a rational ftioction of z, our equation m y becomes 
say 

and here the integers Xj, X^, ,,., X,„ are, each of tliem, equal to or greater than 
Substituting, in the vicinity of a^, the expression 

(i-a.)^Dy=c,0{e^K)A 
provided 



c„(C+«)(d + m-X,.)-Hc„ 



g(«,) 

and the summation for a is for s=l, ..., m except s—r. As X^ is a positive 
integer, and thus is the greater root of tlie modified indicial equation, there is 



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57.] RATIONAL INTEGRALS 173 

one regular integral belonging to the exponent X^, whicii is a constant multiple 
of 

= y, say, where y^ = c^-^Cff, when 5 = A,, 
When we write 

and solve the equations for v^, o.^, ..., we find 
K{&) 

"' /(e+i).../(d+ur"- 

We know (§ 41) that there is a single condition to be satisfied in order that 
the integral belonging to the exponent may be free from logarithms ; as 
f(,6+n) vanishes to the first order for 6=0 when w=Xr, the condition is 

There i^ i jrrespundmg cond tion fjr eaih of the sm^ulirities ajii foi 4, = =o , 
s) thit we haie in + 1 condit una, wtii-h miohe the aibitrary constants 
;! ^n 3 ind t.he positions of the singularities ds well as the assigned 

integers \ ^^ a- t Keeping the latter arlitrary, we aee that there 

m Lst 1 e at least three singularities in the finite part ot the plane when 
there are onlj three we oltiin a limited number of determinations of the 
equation it there aie ''+jo then p elements are left aibitrary among an 
otherwise limited number of determinatiins Df the equation* 

As the oquition is of the setond eider it is possible to plotted otherwise 
Assuming that the integial J" which bekngs to the exponent A^ f the 
singularity a^ la known, and denoting by ^the mtegiil whuh 1 elongs tc the 
exponent cf the same singulaiity welia\e 

rz"- Y"z+{yz'-rz) 2 i^=o, 

so that 



and therefore 



d (Z\ , 1 ™ , ,i -1 



When the right-hand side is expanded in powers of s—a^, the first term 
involves {e—ar)~^~^', that is, the indes is negative. If ^ is to be free from 

Ic^arithms, the term in in this expansion must have its coefficient equal 

to zero — a condition which must be the equivalent of 

• The hypergeometric case indicated in the preceding example is given by 

»,.x.....=x„-i, a. ,,[,-„,). ..if-„j. 

which will be found to satisfy the conditions for a,, ... , a„ given in the text. 



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CHAPTER V. 

Linear Equations of the Second and the Third Orders 
POSSESSING Algebraic Integrals. 

58. The general form of equation, having all its integrals 
regular in the vicinity of each of the singularities (including oo ), 
has been obtained ; in the vicinity of a singularity a, each such 
integral is of the form 

(,^-arH>,+ 'hiog{.~a) + i,,\kgU-a)]' + ...+4,.\hg{z-a:)]-l 
where each of the functions ^n, 0,, ..., 0, is holomorphic at and 
near a. In general, each of the functions is a transcendental 
function in the domain of a: they are polynomials only when 
special relations among the coefficients are satisfied. 

When attention is paid to the aggregate of the integrals so 
obtained, it is to be noted that the branches of a function defined 
by means of an algebraic equation belong to this class. If 
algebraic functions are to be integrals of the differential equation, 
they constitute a special class ; special relations among coefficients 
of the differential equation must then be satisfied, and, it may be, 
special restrictions must be imposed upon its form. Accordingly, 
we proceed to consider those linear equations whose integrals are 
algebraic functions, that is, functions of s defined by an algebraic 
equation between w; and z. It has already been proved (§ 17) 
that each root of such an algebraic equation of any degree in iv 
satisfies a homogeneous linear differential equation, the coefficients 
of which are rational functions of z. If the algebraic equation 
were resoluble into a number of other algebraic equations, neces- 
sarily of lower degree, each such component equation would lead 
to its own differential equation of correspondingly lower order; 
accordingly, we shall assume that the algebraic equation is irre- 



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58.] LINEAR SUBSTITUTIONS 175 

soluble and proceed to consider linear ditFerential equations whose 
integrals are tiie roots of an algebraic equation, , In the most 
genera! case, the degree of the algebraic equation is equal to the 
order of the diffei'entiai equation ; in particular cases (| 17, Note 1) 
it can be greater than the order : and as we seek algebraic inte- 
grals, it may be expected that these particular cases will occur. 

The investigation can be connected with an equivalent problem 
that arises in a different range of ideas. It has been proved that, 
given a fundamental system Wi, w^, ..., w^ of integrals of a linear 
equation of order m,, the effect upon the system, caused by the 
desciiption of a closed path enclosing one or more of the singu- 
larities, is to replace the system by another of the form 



Wm = OmlWi + Oj^W.^ + ... + 0,„mWm I 

say 

(iU]' Wm') — S{Wi, ...,w,„), 

where S denotes a linear substitution. By making the inde- 
pendent variable describe an unlimited number of contours any 
number of times, we may obtain an unlimited number of linear 
substitutions ; and so each integral could, in that case, be 
made to have an unlimited number of values. If, however, the 
fundamental system is equivalent to the m roots of an algebraic 
equation, then each of the integrals can acquire only a limited 
number of values at a point which are distinct from one another: 
that is, there can be only a limited number of substitutions in 
the aggregate. When therefore we know all the groups of linear 
substitutions in m variables which are of finite order, only those 
linear differential equations which possess such groups need be 
considered. Accordingly, if we proceed by this method, it is 
necessary to construct the finite groups of linear substitutions. 
Further, it is clear that the investigation can be associated 
with the theory of invariantive forms ; for the relations between 
w/, ..., Wju' and w,, ..., w^ constitute a linear transformation of 
the type under which these invariantive forms persist. Indeed, 
it was by this association with binary, ternary, and quaternary 
forms that the earliest results, relating to linear equations of the 
orders two, three, and four, were obtained. Some brief indications 
of this method will be given later (^ 69 — 72). 



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EQUATIONS OF THE SECOND ORDER [59, 



Klein's Method tor Equations of the Second Order. 

59. The determination of linear equations of the second order, 
whose integrals are everywhere algebraic, is effected by Klein*, by 
a special method that associates it with the finite groups of linear 
substitutions of two homogeneous variables. 

Let w, and w, denote a fundamental system of integrals for 
the differential equation ; and let 

Ifi = awi + jSws, JTs = 7W, + Swj , 
be any one of the linear substitutions, representing the change 
made upon the fundamental system by the description of a closed 
path. Then taking 

Wj' 

the quotient of two algebraic integrals, so that s itself is an 
algebraic function, we have 

W.^ys + B' 
thus s is subject to a homographic substitution. Accordingly, 
the determination of the finite groups of linear substitutions in 
the present case is effectively the determination of the finite 
groups of homographic substitutions. 

Let any such group containing N substitutions be represented 

by 

t.W. t.(«) +«(«). 

and let t^,, (s) = s, the identical substitution : every possible com- 
bination of these substitutions can be expressed as some one of 
the members of the group. Take a couple of arbitrary constants 
a and b, subject solely to the negative restrictions that a is not 
equal to ■<frr(b) and b is not equal to -^((a), for any of the values 
0, 1, .,., N ~1 of r and of s; and form the equation 

^a(s) — a ifrj (s) - a t^tj^., (s ) - a _ „ 

Ms)~b■^ir,{s)-b ir^-d'^)-b 

* Math. Ann., t. ii (1877), pp. 115—110, ih., t. xu (1877), pp. 107—179; 
Varlemngen Hberdai Ikosaeder, {LeipKig, Teubaer, 1S84), pp. 113—123. 



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59.] wrrn algebraic integrals 177 

which is an algebraic equation of degree W^ in s. It is unaltered 
when s is submitted to any of the substitutions of the group ; for 
such a substitution only effects a permutation of the various N 
fractions on the left-hand side among one another. Hence, if any 
root s be known, all the N roots caa be derived from it by 
submitting it to the JV substitutions of the group in turn. 

For quite general values of X, the iV roots of the equation are 
distinct; but it can happen that, for particular values of X, a 
repeated root arises, of multiplicity v. From the nature of the 
equation in relation to the group of substitutions, it follows that 
each distinct root is of multiplicity c, so that there are N—v 
distinct roots. To consider the effect of this property of the 
equation, let the latter be changed so that the numerator and 
denominator are mulfcipKed by the denominators of i|'i(s), ..., 
^fr^^l(s). It thus Can be expressed in the form 

where G (s, a) is a polynomial in s of degree N", the coefficients 
being functions of a, and G {s, J) is a similar polynomial, its 
coefficients being the same functions of b. Let X, be a value 
of X, such that 5 = <t, is a root of multiplicity v^ when X = X^ ; 
then the equation 

G(s,a) 0{a,,a) _ 

Q\s, h) G(cr„ b) ' 

N 
has — roots each of multiplicity v^ when X =X,. But each such 

root is a root of multiplicity iij ~ 1 of the equation 

d [ gfoo) gfa, ■») )_■ 
'<u\e{s.bj G(r„6)( ■ 

that is, of the equation 

AW = fl(»..)^-«it^-<J(..)«(''-A) = 0; 



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178 Klein's method for [59. 

of fclie roots of this derived equation. Moreover, we then have 



G(s,b}G{<y„b) 



= X-X,. 



Let X^ be another vahie of X, suoh that s = cr^ is a root of the 
equation of multiplicity j-j when X = X^. A precisely similar 
argument shews that each distinct root of the equation is of 
multiplicity v^; that there are N-^v^ distinct roots; that each 
such root is of multiplicity v^—l for the equation A (s) = ; that 
these roots account for 

f(.-i) 

of the roots of the derived equation ; and that we have 

O/' -r x 

N 
where 'I's is a polynomial in s of deg;ree ■— . 

Proceeding in this way with the various values of X that lead 
to multiple roots of the initial equation, we shall exhaust all the 
roots of the equation A (s) = 0. The degree of A (s) is 2jV — 2 ; 
for if 

G(s.a) = s^Ma) + s''-'Ma)+ ..., 
then 

B{s.b)-^/.(b) + s''-'Mb)+...; 
and therefore 

4(.)-»'"l/.(«)/>(6)-/.(4)/.(«)) + -. 
But taking account of the roots of A (s) = 0, as associated with the 
multiple roots of the original equation for the respective values of 
X, we see that its d 



and therefore 






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59.] EQUATIONS OF THE SECOND OKDEIi 179 

Each of the integers v is equal to or greater than 2, so that each 

of the quantities 1 is equal to or greater than J. Hence the 

smallest number of different integers i- is two ; if there were only 
one, the left-hand side would be < 1, while the right-hand side is 
> 1. The largest number of different integers p is three ; if there 
were four or more, the left-hand side would be equal to or greater 
than 2, while the right-hand side is less than 2. 

In the first place, let there be only two integers, y, and v^ ; 
then 

1 J. __2 

From the nature of the case, v, < iV, v^ ^ JV, so that 

hence the only possible solution is 

:,. = JV, v, = N; (I), 

and N is an undetermined integer. 

In the next place, let there be three integers, v^, v^, v, : then 
111,2 

Vi Vi 1-3 JS 

At least one of the integers v must be 2 : for if each of these 
integers were ^ 3, the left-hand side would be < 1, while the 
right-hand side is >1, as iV is a finite integer. 

Taking j',= 2, we have 

11,2 

Another of the integers v may be 2. Let it be v^\ then N = ^V3, 
and we have the solution 

v, = 2, v,^2, v, = n, JV=27i, (II), 

where n is an undetermined integer. 

If neither of the integers p^ and v^ be 2, one of them 
must be 3 ; for if each of them were ^ 4, then - + —^^, and so 



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180 ALGEBRAIC INTEGBALS AND [59. 

could certainly not be equal to ^ + -^ . Taking v^ — 3, we have 



jV 



N' 



so that Cj < 6 : thus possible values of v^ ai-e 3, 4, 5. The solu- 
tions are 

j,, = 2, r, = 3, v^=% N=12, (Ill), 

„j=2, ^3 = 3, i; = i, N=2i, (IV), 

v, = 2, v^^S, v,= 5, iV=60 (V). 

60. The finite groups are thus known ; the corresponding 
equations in s are required. The solutions will be taken in 
order. 

I. Instead of X, we take a quantity Z, defined by the rela- 
tion 

, x-x, 

so that Z=0 gives X — X^, that is, gives s = Si, a root repeated 
N times, and Z—xi gives X = X^, that is, gives s = s^, a root 
repeated N times. We have 

J V ('-'■)•• 

' e(s, l>)G(j„i)' 



X-X,~^i^,: 



and therefore 



absorbing the constant (? (Sj, b)-r-G (s^, b) into the variable Z. 

11, III, IV, V. These cases are of the same general form. 
Instead of X, we take a quantity Z, defined by the relation 



X-X, X,-X,' 

then 7=0 gives j: = X„ 2.1 gives X = X,, 2=« givesX-X,, 
and thus 
Z:Z-1 : 1 

= (X - X,) (X, - X.) : (X - X,) (X, - X.) : (X - X.) (X, - X,). 



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60.] POLYHEDRAL FUNCTIONS 181 

But 

' G(s,b)G(aub)' 

Y Y - '^°^' 

^ G{s,b)G(^,,b)' 

^~-^'^G{s,b)G(^,,by 
and therefore 

^ : 2-1 : 1 = ^*/^(s) : B<t',-'(s) : ^."'(s). 

where A and B are constants which, if we please, may be absorbed 
into the functions 3>s and <I», respectively. 

Now these groups are the groups that occur in connection 
with the polyhedral functions* : and the polyhedral functions can 
be associated with the conformal repreaentation-|-, upon a half-plane, 
of a triangle, bounded by three circular arcs and having angles 

equal to - , - , - , The analytical results connected with these 

investigations can be at once applied to the present problem. 
Denoting derivatives of Z with regard to s by Z', Z" , Z"', ..., we 
have (T. F., § 275) 



Z'\Z' ^'\Z')\ ^ 



or, taking account of the properties^ of the Schwarzian derivative, 
we have 

' ■ ' Z- '^ (2-1)" Z{Z-l) 

The forms of the functions for the various cases II, III, IV, V 



for II, 

* T. P„ g§ 276—279, 300—302. 

t T. F., %% 27*. 275. 

+ See Ex. 3, § 62, of my Treatise on Differential Equations. 



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182 ALGEBRAIC INTEGRALS OF [60. 

for III, 

Z:Z-\ :1 

= (s* + 2sV3 - 1)= : lav's sHs* + 1)^ : (s^ - 2s^V3 - 1)= ; 
for IV, 

Z:Z~l :1 

= (s^ + 14s* + l)»:(s^=--33s'-33s'+l)^: 1085^(5'- 1)= ; 
and for V, 

Z.Z-1 : 1 = (s^~~ 228s" + 4-9is"' + 228s' + If 

: {s*=+ 1 + 522s''(s™ - 1) - 10005s'Hs"' + l)p 
;-1728s«(s" + lls°-l)=. 

These results* can be obtained by purely algebraic processes, 
from the properties of finite groups proved by Gordanf. 

61. These results can be applied at once to the determination 
of linear equations of the second order 

<Pw dw 

all the integrals of which are algebraic. Denoting the quotient 
of two integrals Wi and w^ by s, we have§ 

w =5'"*se"*^'^ w =s''^e'^'^'^, w.,s = w 

say. As all integrals are to be algebraic, it follows that s and 
s'-S are algebraic ; accordingly, fpdz must be the logarithm of an 
algebraic fv/nction, which is a Jirst condition. Further, in the 
equations under consideration, both p and q (and therefore also 
2/) are rational functions of z ; and therefore 

[s, z] = rational function of z, 

* They are sliglitly changed front the forms in % 302, g 278 {I.e.) ; the ciiange is 
made, eo aa to associate the indices v.^, t^, y.j with the values Z = 0, Z — 1. Z = rrj 
respeo lively. 

t Math. Ann., t. m (1877), pp. 23^6. See also Cajley's memoir, "On the 
achwaraian derivative and the polyhedral functions," Coll. Math. Papers, t. Si, 
pp. 148—216. 

g See my Treatise on Differential Equaiions, §g SI, G2. 



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61.] EQUATIONS OF THE SECOND ORDER 183 

and the quantity s is subject to the transformation of the finite 
group. Now we have seen that 

\'.^ F~+ ik-if + Wz^ ' 

in cases II, III, IV, V ; and for case I, it is easy to verify directly 
that 



From the properties of the Schwarzian derivative, we have 

hence, taking account of the particular form of [s, Z] which is 
actually known, and of the generic form of [s, z] which ia required, 
we see that, in order to satisfy the conditions, we must have 

Z=R{z), 

where K is a rational function of z. Conversely, the conditions will 
I if .Z' is any rational function of z. Accordingly, the 
<,l equation oftJie second order viust have the coefficient of 
It/ in the form 



where u is om algebraic function of z ; and its invariant I{z), 
which is q — \P' — h'j^' '"''"^* ^s of the form 



1--. — + ^*^- 

^{Z-\Y^ Z{Z^i) 



\\Z,z}, 



where Z is any rational function of z ; the integers v-^, v^, va in the 
first form are the integers of the finite groups in cases II, III, IV, 



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184 MODE OF OBTAINING [61. 

V ; and N in the second form is an integer. When these con- 
ditions are Batisfied, the integrals are given by 

where, for the first form, s is determined in terms of Z, the 
rational function of z, by the equations at the end of § 60 ; and for 
the second form. 



Construction of an Integral, when it is Algebeaic. 

62. The preceding investigation is adequate for the general 
construction of linear equations of the second order which are 
integrable algebraically ; there still remains the question of 
determining whether any particular given equation satisfies 
the test. 

When the equation is of the form 
d'vj dw 

inspection of the form of p at once determines whether it satisfies 
the condition which governs it specially. Assuming this con- 
dition to be satisfied, we construct the invariant I{z) of the 
equation, where 

and then, if the original equation is algebraically integrable, we 
must also have 



(2-1)' Z{Z-1) 



/W-l^^TT, +11^.^ 



©■-il^^!. 



where ^ is a rational function of 2, and the integers c,, v^, vs 
belong to one of four definite systems. 

It may happen that the identification is easy, because Z has 
some simple value; the simplest of all is, of course, given by 



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62.] ALGEBRAIC INTEGEAL9 185 

Z= z. When the identification is not thus obvious, it is desirable 
to have a method of constnicting the rational function Z if it 
exists ; when it has been constructed, the further id entiii cation is 
only a matter of comparing coefficients. Should this identification 
be completely effected, then the integration of the equation is 
given by the results of § 60, 

Such a method is given by Klein*, who uses for the purpose a 
comparison of those terms on the two sides, which are connected 
with the poles and have the highest negative index. A rational 
function is detenninate save as to a constant factor, when its 
zeros, its poles in the iinite part of the plane, and their respective 
multiplicities, all are known ; and this constant factor is determ- 
inate, when the value of the rational function is known for any 
other value of the variable. Accordingly, let a denote a zero of ^ 
of multiplicity a, and so for all the zeros ; let c denote a pole of Z 
{and therefore also of Z— 1) of multiplicity y, and so for all the 
poles ; and let h denote a zero ot Z — 1 of multiplicity ^, and so 
for all its zeros : then 

U{z-ay U{b-c)y 

n{b~ay ■ n(z-c)y' 
where the multiplicity ^ of 6 is not used directly in the ex- 
pression. 

Consider now the right-hand side of the expression for I (i). 
In the vicinity of a, we have 

where t7 is a regular function oi 2~a, not vanisiiing wlien s = a; 
so tliat 

IdZ 'J „, . 

^JJ-^3i + -'*<'-<■>. 
and 

the unexpressed terms in [Z, z] having exponents greater than — 2. 
In the vicinity of c, we have 

Z=(z-cyiV, Z-l=(z-c)-iV,. 

" Math. Aim., t. sii (1877), pp. 173—176; the espoaitiou given in the test 
does not follow his eiactlj, as he transforms the equation 90 aa to secure tliat 
s = o: ia an ordinary point. 



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186 CONSTRUCTION OF AK [62. 

where V and V^ are regular functions of 2 — c, not vanishing when 
z=c; thus 

IdZ -7 „, 

1_ dZ -, 
Z-ldz ^ z- 



_fS,{^-c), 
1^. ^ 



-e)" ■■■' 

the unexpressed terms in \Z, s] having exponents greater than — 2. 
In the vicinity of 6, we have 

2-l-(»-6)»If, 
where TT is a regular function oi s — h, not vanishing when z=h\ 
Ho that 

Z-\d-^ z-b* ^ °'' 

the unexpressed terms in [Z, z\ having exponents greater than — 2. 

We thus have taken account of all the highest terras with 
negative indices which arise through zeros or poles of Z and Z—1. 
On account of the form of [Z, z], which is 

Z' ^\Z') ' 
it is necessary to take account of the poles and the zeros of Z'. 
As Z is rational, all its poles are poles of Z' and the latter has no 
others; so that, on this score, no new terms arise, A repeated 
zero of ^ is a zero of Z', and all these have heen taken into 
account; likewise for a repeated zero of Z—1. Hence we need 
only consider those roots of Z', which are not repeated roots of Z 
or of if — 1 ; let such an one be t, of multiplicity t, so that 

z.(z-tYqu-e,. 

where Q is a regular function of ^ — (, not vanishing when s = t; 
then 

the unexpressed terms io [Z, s] having exponents greater than — 2. 



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62.] ALGEBRAIC INTEGRAL 187 

Gathering together the terms with the largest negative index, 
we have, for Cases II, III, IV, V, 

(s- ay (s - by {z - cf {z - ty 

where the unexpressed terms have integer exponents greater 
than — 2 ; and in this expression the significance of a, b, c, for the 
construction of Z, must be borne in mind. Actual comparison 
with the form of / {z) then gives indications as to which set of 
values of II,, Vi, Vj must he chosen, and determines the values of 
a, ^, 7- The construction of Z is then possible and, Z being known, 
the complete identification of the right-hand side with the known 
value of /(if) is merely a matter of numerical calculation. 

For Case I, we have 

and the method of proceeding is the same as before. 

In particular instances, it may happen that no terms of the 
type 

tr + JT- 
(s-tf 

occur : Z' then contains no roots other than the repeated roots of 
Z and Z — 1. An example is given by 

^- 4^' 

Further, it may happen that a= v-^, or ^ = vi, or y^Vsi so 
that the corresponding value of z, viz. a, b, or c, is then not a 
singularity of the differential equation. And, in particular, if 
if = CO is not a singularity of the differential equation and there- 
fore also not a singularity of the integral, then, if the equation be 
integrable algebraically, the numerator of the rational function Z 
is a polynomial in z of the same degree as the denominator*. 

" Thin form of equation is discoased by Klein in the memoir already quoted 
(note, p. 186) ; reference should he made to it tor further developments. 



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188 

M:^. 1. The equatio] 






blo algebraically, For 






l% = l(l-^)> whence J.2 = 2; 



We thus have an instance of case II, when a = 2. All the conditions a 
satisfied : and thus (§ 60) the integrals of the equation are givon by 






fie. 3. Construct a linear differential equation of the second order in its 
normal form, such that the quotient s of two of its solutions is given by 

108s»(si-l)a 43 ' 

Sx. 3. Consider the equation 

We have 

. ^ 2s>-8s=-153a-82 + 2 (a-l) ' 

the terms indicated constituting all the infinities of /(s) of the second order. 

First, it is clear that there is only one root of Z' other than repeated roots 
of Z and Z— 1 ; it is characterised by 



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62.] Klein's method 189 

If it wereposBibly an instance of case II with m = 3, then we must have 
^^l_^^^=^,sothati., = 2, 3 = 1, h^i, 

i{^-^^-^> >',-3,y= I, <: = (>, 

i(l-5) = 3^. ., = 3,y = 2,«=-l; 

and therefore 

with the condition that 2—1 when z = h = i, so that A=i. But then 

shewing that Z' docs not possess a root z = l= 1 ; hence the example is not an 
instance of case II. 

If therefore the equation is algebraically integrahle, it must bo an instance 
of case III. We must have therefore 

i{ 1 — ^)=A^i whence ff=l, b=i, 



and then, either 



Q = l, a = Q, y = % c=-l; 



Taking the former, we have 

from the poles and zeros ai Z\ s& Z= 1, when s = i, we have A = 2, bo that 

SO that Z- I has the roots 2 = 8, s= - i; hut 



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shewing that Z' does not possess the f. 
values is not possible. 

Taking the latter, we have 



; and thus the first assignment of 



■sof^; asZ=l when s-i, 



e have A =^, and then 



so that Z—\ has 2=i,z— -ifor roots, and Z" has 2=1 for a root. 

The preliroinavy conditions are thus satisfied ; it is easy to verify that 
this value of Z gives the coinpleto value of /(:). Hence, after the results of 
§ 60, the intf^ral of the differential equation is given by the equations 

^si + 2aV3-l Y _ {a-VVf 
\} 2s ' 



algebraically integrable. 




^_ +<S-i*.)s-A-o, 

and bt n th ntegrals 



where f=(i-ai)(^-a2)(^-''s),^dX,=i{a-a),X,=iO-^),),3=My-/}i 
discuss the possibilities of algebraic integrability for the values 

\^h ^2 = f. \ = \- 
In particular, shew that, if %= - 1, «3=0, then 



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EQUATIONS OF THE THiRD ORDER 



Equations of the Thied Order with Algebraic Integrals. 

63, When we pass to the considGration of hnear equations of 
order higher than the second which are algebraically integrable, 
the discussion can be initiated in the same way as for equations 
of the second order ; but the detailed devdopment proves to be 
exceedingly laborious, and it has not been fully completed for 
each case. Only a sketch will here be given 

Dealing in particular with the linear equation of the third 
order, we take it in the form 



where p, q, r are rational functions of s, subject to the limitations 
imposed by the regularity of the integrals in the vicinity of eafih 
singularity (x included). If w^, w^, w, denote three i 
independent integrals, we have (§ 9) 



BO that, as Wj, Wa, Ws are algebraic functions of s, it follows that p, 
a rational function of z, must be of the form 

where m is an algebraic function of s. This is a first condition : it 
is the same as for the equation of the second order (§ 61): and it 
is easily obtained as a universal condition attaching to any linear 
equation which is algebraically integrable. 

Now substitute for v) by the relation 

and let y^, y^, y^ denote the three integrals corresponding to 
M>i, Wa, Misi owing to the character of p and the functional 
character of the integrals w, the integrals y are also algebraic 
functions of z. Thus the equation in y, being 



/" + 3Qy + ii-0, 



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192 EQUATIONS OF THE THIRD ORDER [63. 

where 

Q = q-p'-p' I 

R ^ r - Spq -i- ^p' - p" ) ' 
is to be algebraically integrable. Denoting by s and t the 
quotients of two integrals by a third, we have 

The quantities s and t are algebraic functions of z for equations 
of the class under consideration. 

The effect upon a fundamental system, when the independent 
variable describes a circuit enclosing one or more of the singulari- 
ties, is represented by relations of the form 

F, = a y, + b j/a + c 

Fj = «' ^1 + b' y^ -+ c' 

7a = (t"l/i + &>, + c"(/3 J 
If S and T denote the corresponding integral-quotients, then 
„ ^ a' + b's + c't „ ^ a" + b"s + o "t 
a + bs + ci ' a + bs + ct 

Now if the equation is integrable algebraically, there can exist 
only a limited number of different sets of values of the integrals ; 
so that the number of sets Y,, ¥^, Y^ is finite, and the number of 
simultaneous values of S and T is finite. If then we know all the 
homogeneous linear gioups m three variables, or (what is the 
same thing) all the lineo-lineai groups in two variables, which are 
finite, then each such finite group determines its set of values of 
Yi, Y,, Fa and the set of values of S and T, and so it determines 
a linear equation the integrals of which are algebraic: and con- 
versely, each such linear equation is characterised by a finite 
group. 

64. In order to utilise the method for the present purpose 
on the lines adopted for the equation of the second order, it is 
necessary to deduce from the differential equation certain differen- 
tial invariants involving s and *, these invariants being expressed 
in terms of Q and E. This can be done in two ways. It is clear 
that, as s implicitly contains five arbitrary constants, it satisfies a 
differentia! equation of order five ; and that, as ( is of the same 
functional form as s, it satisfies the same differential equation. 



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64.] WITH ALGEBRAIC INTEGRALS 193 

On the other hand, as s and ( combined contain eight arbitrary 
constants implicitly, it may be expected that the two differential 
equations, which they satisfy and which wili involve both of them, 
will be each of the fourth order or will he equivalent to two of 
the fourth order. The single equation is, for some purposes, the 
more important in the formal theory of the hnear equation, which 
will be left undiscussed ; for the present purpose, the two equa- 
tions prove to be the more important. Accordingly, we substitute 



sy, for 1/2, and tz/, for y^, 



1 the equation 



I integral of this equation, > 



whence, remembering that y^ 
have 

3s'y," + 3fi'>/ + {SQs' + s'") y, = 0| 

3(>," + 2t'%' + (SQt' + t'") !/, = 0) 

Differentiating each of these once, and substituting for y,'" from 

the linear equation which it satisfies, we have 

Qs'%" + (4s'" ~ 6Qs') y/ + [s"" + SQs" + 3 (Q' - R) s'] 2/1 = 0) 
Qt"y," + (W" - GQt') y: + [*"" + 3Qt" + 3 (Q' - ii) t'} y, = 0\' 

so that there are four equations, linear and homogeneous in the 
quantities y", y-[, y,. When the ratios of y" : yl : y^ are eliminated 
from the first pair and the first of the second pair, we have 

-3(ii-Q') 



and when the same ratios a 
pair and the second of the se 



i likewise eliminated from the first 
>nd pair, we have 



("", «'" 


6!" 


-3Q 


»'", 3." 


3«' 




1'", 3i" 


31' 





-3(E-Q') 



('", 3f", 3(' 

These, in fact, are the two equatio 
satisfied by s and (. 

F. IV. 



0. 



f , , 

s"\ 35", 3s' 

*"', 3(", 3(' 

, each of the fourth order, 

13 



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194 INVARIANTS FOE AN EQUATION [64 

Suppose now that two solutions (other than the trivial solu- 
tions, s = constant, ( = constant) are known, say 

Solving the first pair of the foregoing equations for y^ : y,, 
we have 

3 (aW - o-V") y,' + (^'"t - a'r'") y, = 0, 

and therefore 

neglecting an arhitrary constant arising as a factor on the right- 
hand side. Hence a fundamental system of integrals of the 
original equation is 



{</'t 






n~ 



tWt'-c 



■i. 



or the original equation can be integrated if two particular 
solutions of the equations in s and t are known. 

65, Moreover, from the source of the two equations which serve 
to determine s and i, it is to be expected that, when the above 
two (being any two) particular solutions s— a-, t = T, are known, 
the complete primitive of the two equations is 



'/ + b'lT + c't 

ll+b(T+ GT 



t = 



where the constants a, h, c, a', V, c', a", b", c" are arbitrary so fe,r 
as those two equations are concerned. This result can be stated 
in a different form. The two equations in question can be written 

As"" + iBs"' + 6Gs" - SQ (As" + 2Bs') - 3 (-K - Q') As' = 0, 
Alf'" + 4Bt"' + 6Gt" - SQ (At" + 2Bt') -S{R- Q') At' = 0, 
where A, B,G are the three determinants in 

I, t"'. 3t", W I 



Its — s"" t - 



I ii" t" - 



s"t'". 



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65.] OF THE THIRD ORDER 195 

SO that 

^ = 9mi, £ = -3%, 6' = 3^,; 

then solving the preceding equations for Q and for R — Q' in turn, 
we find 

3e=!^'i-^-ig)'-/(M,.) ) 

and ' ' I, 

-2T(-S-Q-) = 9°'-6 °-'°'t*°' - + sf-)'-f(».'.^) 
Ill W[^ \wi/ ; 

say. The latter equations may he regarded as the equivalent of 

the two equations, which have been solved ; and therefore we may 

expect that 

/ / «' + f>'s + c't a"+b"3+c"t ^]^j-(^ f ^-j 

r faf + b's + c't a" + 6"s +c"t \ r , ^ , 
\a + bs + ct a + bs + ct J \ ■ ■ " 
the actual verification, which is comparatively simple, is left as an 
esercisa Clearly these are generalisations of the property of the 
Schwarzian derivative, represented by 



[cs + d j 
The two invariant functions / and J were first indicated* by 
Painlev^ ; they subsequently were simpUfied to a form, which is the 
equivalent of the above, by Boulanger-f-. 

The invariance of the functions / and J, as indicated, exists 
for lineo-linear transformation of s and t. There is also an 
invariance for any transformation of the independent variable z ; 
for we easily find the equations 

I(s, t, z)^I{s, t, Z)Z-'+2{Z, z], 
J(s, t, z) = J(s, t, Z) Z''~9I (s, t, Z) Z'Z" ~ 9 ^ (^. ^1- 
where Z is any function of z. Also 

^I'(3,t,2 

' Comptes Efndus, I. oiv (1887), p, 1830. 

+ See his Thftse, Contribution h I'elude des ^q'uatiimB differentielies Un$a 
it lumioginei intSgrabUs algSbriquement, [Paris, Gautliier-Villarfl, 1897). 

13—2 



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196 INVARIANTS AND [65. 

and therefore 

j(s, t, z)+^r{s, t. .) = [J(s, (, Z) + ^r{s. t, Z)-\7i\ 

is an invariant for any change of the independent variable s. 
Dropping a numerical constant, this is the function 



which is the known Lagnerre invariant in the formal theory ; that 
is*, if the equation 

be transformed, by the relation 



--(sr 



to the form 
then 



As the transformation 



--»f=(^.-ii)(SJ' 

'dZ\-^ 



1-' yii) 

leaves the quotient of two integrals transformed only as by a 
lineo-linear substitution, it follows that the preceding function, say 

L (s, t, s) = J is, t, z) + f 7' (s, t. z), 
is unchanged by lineo-linear transformations effected on s, t; 
also, except as to a factor Z'', it is unchanged by transformation 
effected on the independent variable. Now 



30 that we have 



" See a paper by the author, Phil. Tram. (1888), pp. 383, 390, Lagnerre's 
invariant was first aanounoed in two notoe, Comptee Hendus, t. Lsixviii (1879), 
pp. 116—119, 224—227. 



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65.] FINITE GROUPS 197 

which is the full expression of Laguerre's invariant in terms of 
the derivatives of s and (, 

66. The next stage is to associate these invariants with the 
algebraic equations in two variables, which admit of one or other 
of the finite groups. These groups have been obtained by Jordan* 
and Valentinerf ; and references to other writers are given by 
Boulangeri- A method of using the results is outlined by Pain- 
lev^g as follows. 

Let (s, t), 1^ (s, () denote two irreducible invariant functions 
of a finite group of order JV; the functions are given by Klein|| for 
the group of order 168, and by Eoiilanger (I.e.) for the group of 
order 216. As these functions are invariable for each substitution 
of the group, and as s, t are algebraic fimctions of s, it follows 
that and -i^ are rational functions of s, say 

</,{.,() = * (4 f(s,t)=^^{^). 

Conversely, taking * and ^ to be arbitrary rational functions of z, 
these two equations give rise to N sets of simultaneous values 
of s and t as algebraic functions of z ; and if any one set of 
values be represented by <t, r, all the others are obtained on 
transforming a- and t by all the iV - 1 substitutions of the group 
other than the identical substitution. These two equations are 
used to obtain the first four derivatives of s and ( with regard to s ; 
and with these derivatives, the two invariants 

I(s,t,^). J{s,t,z) 

are constructed. The functions so formed involve derivatives of 
<!> and ^ ; and the coefficients of these quantities are rational in 
the derivatives of <^ (s, t) and i^ (s, t). As 7 and J are invariantive 
for the group, the coeiEcients specified are rational functions of s 
and t, which must be invariantive for the group and are therefore 
rationally expressible in terms of ^ and t/t, that is, in terms of <& 

• Crelk, t. Lxxxiv il878), pp. 89—215 ; AUi delta Jl. Accad. di Napoli, t. viii 
(1879), No. II. 

t Kj^b. Videmk. SeUk. Skr., 6 E., t. v (1889), pp. 61—235. 

X In the Ttese, already cited on p. 19S, note. 

% Comptei EendMS, t. civ (1887), pp. 1829—1832, ib, t. cv (1887], pp. 58—61. 

II Math. Ann., t. xv (1879), pp. 265-267. 



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198 ALGEBRAIC [66. 

and ^, Thus I(s, t, e) and J{s, t, z) would be expressed as 
rational functions of s. Accordingly, taking 

3Q = /(s,(,2), 

R = hI'{s,t.z)-i,J{s.t,z), 

we have the differential equation 

y'" + my' + Ry = 0. 

The earlier investigations shewed that its integi-als are expressible 
in terms of s, t, and their derivatives ; and we thus have a method 
of constructing all the linear differential equations of the third 
order which are integrable algebraically. There is a double 
arbitrary element for each group, viz. the arbitrary forms of the 
rational functions $ and 'P ; and there is a limited number of 
groups, 

67. While this outline is simple enough in general descrip- 
tion, the application to particular cases requires extremely elabo- 
rate calculations. These have beeti effected by Boulanger for the 
group of order 216 ; they do not appear to have been yet effected 
for any one of the other groups. As, however, the enumeration 
of the finite groups in two quantities s aud ( is complete, the 
subject offers an. interesting, if a laborious, field of investigation. 

In the absence of the complete table of equations, for all the 
finite groups and for two arbitrarily assumed functions ^ and "^j 
it is not possible to use a method, analogous to that of § 62, to 
determine whether a given equation of the third order is algebrai- 
cally integrable or not; it is not even possible to recognise to 
which of the groups it would belong if it were algebraically 
integrable. Indications of two general methods of procedure have 
been given by Painlev4 and have been developed to some extent 
by Boulanger; but the methods, while general in description, 
suffer from the same kind of difficulty as the method indicated 
for the construction of the equations, for the calculations are 
exceedingly laborious. We have seen that, if two particular 
values of s and (, say t aud t, are known, then an integral of 
the differential equation is given by 

J/ = (o-'V---tV')-*. 



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67.] INTEGRALS 

Hence, if we take 



so that the number of values, which u can acquire, is equal to M 
or to a Hubmultiple of N, where JV is the order of the associated 
group; let the number of values be n. Now if y ia algebraic, 
every zero of y and every infinity of y are of a finite order, which 
is commensurable in every instance ; and therefore all the infinities 
of u are simple poles with commensurable residues. Substituting 
for u in the ecjuation 

y'" + ^Qy +Ry = 0, 
we find 

u" + Suu' + w^ + SQu + R=0, 

a non-linear equation of the second order satisfied by u. This 
equation renders it possible to test the character of the poles and 
the residues of u. If these are of the appropriate type, then the 
equation is satisfied by a relation of the form 

where A^, Ai, ,.., An are polynomials in s, and A^ is the product 
of the factors corresponding to the poles of a. Then there is the 
further test that this algebraic function u must be such that 

is algebraic. Manifestly, the calculations will generally be too 
elaborate to make the method eifective in practice. 



Equations of tub Fourth Order. 

68. As pointed out* by Painlev^ the processes just indicated 
can formally be applied to linear equations of any order: but of 
course, if any advance towards final conditions is to he made, it is 
necessary to know all the finite lineo-linear groups of transforma- 
tions in a number of variables less by one than the order of the 

' Compls Eendua, i. cv (1887), p. 59. 



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200 EQUATIONS OF THE FOURTH ORDER [68. 

equation. Towards this enumeration of groups in three varia- 
bles, which are associated with the linear equation of the fourth 
order, Jordan* has constructed a characteristic numerical equation 
which, when completely resolved, would indicate the order and 
the composition of each such group : but the resolution is exceed- 
ingly long and, owing to the number of cases that must be 
considered, it has not been completed. In these circumstances, 
no detailed results of a final critical character can be obtained for 
an equation of the fourth order or of any higher order : the only 
results obtainable are of a general character, and arise through the 
association of groups in general with linear equations. 

The equation of the fourth order, which may be written 
w"" + 4pMi"' + 6}W!" + 4rw' + swi = 0, 
can be transformed by 

^g/p<i= ^ y 

into 

f" + eQy" + iUy' + 8^0. 

We denote a system of four integrals by j(i, (/,, y,, y^, and we 
introduce three quotients s, t, u, such that 

then s, (, u are simultaneous solutions of three equations of the 
fifth order in the derivatives. If a-, r, v are a special set of 
solutions, then 



yi = 



-i 



yi = yi<y, Vi = y^t, y^ = ^if ■ 

The complete primitive of the three equations is of the form 

s _ _ _t u 

a' + b'a- + c't + rf^ ~ tt" -i- 6'V 4- c"t + d'V ~ a'" + h"'<T + d"r + 



' Atti della B. Accad. di Napoli, t, viii (1870). No. 11, p. 25; instead of 
dealing with lineo-luiear traasformations in three vacia.bles, Jordan deals ivith 
homogeneouB linear aubstitutioiiB in four variables. 



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68.] WITH ALGEBBAIC INTEGRALS 201 

There are three functions of the derivatives of s, t, u, with regard 
to z, which are invariantive for substitutions such as the precodiug 
relations expressing s, (, u, in terms of a, t, v; and they are 
equal to 

If the determinants 

S ± {8"'t"u'\ % ± (s"t"u'), S ± {s''t"'u'), S + {s't'V) 
be denoted by p, pi, p.^, p^ respectively, then 

say ; if, in addition, the determinants 

£ ± (s'Tu"), S ± (s-'f'u') 
be denoted by p^ and p^ respectively, then 

say ; and if the determinant 2 ± (s''s"'s") be denoted by p^, then 



S-so.= -& ,„„„,„„„ 



«-3;-8e-=f;-i87»*+2W. 



say. The three quantities I^ (s, t, u, z), /g (s, (, u, z), I^ (s, t, u, z) 
are unchanged when lineo-linear substitutions are effected on 
s, t, u; and the combinations 

/.+ 2/,--l/,"-A/,', 

are also unchanged, except as to a power of Z', when e is replaced 
by Z, any function of z. 

The proofs of these various statements are left as exercises. 



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202 algebraic integrals and [69. 

Equations, having Algebraic Integrals, associated with 
Homogeneous Forms. 

69. It has already {§ 58) been stated that the discussion of 
the equations, which have algebraic integrals, has been associated 
with the theory of homogeneous forms : the association can be 
seen to occur as follows. 

Using the preceding notation of §§ 63 — 66 for the quantities 
connected with any linear equation of the third order, we denote 
by s and ( the quotients of any two by the third out of any three 
linearly independent integrals of the equation 

If, then, all the integrals of this equation are algebraic, both s 
and t are algebraic functions of z ; they may therefore be 
regarded as determined, in the most general case, by a couple of 
distinct algebraic equations, say 

/.(»,(, »).0, /,(s,i, «) = 0, 
or by 

9,{s.^) = 0, g,{t,z) = {). 

Eliminating z between the pair of equations in whichever form 
they are taken, we obtain a relation of the type 

i^i,(s, = 0, 
where Fg is a non- homogeneous polynomial in s and t, because it 
is the eliminant of two polynomials. Replacing s and i by jij -^ y-i 
and y^-i-yi respectively, and multiplying by the proper power of 
3/1 to free the equation from fractions, we have 

^ (yu y2, y^ = ^y, 

where f is a homogeneous polynomial in its arguments or, in 
other phrase, is a ternary form in 3/,, y,, y,. 

Further, the above form of equation is obtained from 



dht) , „ (iHv , „ dw , „ 

■j T + 3jo , T- +3q-:j- + rw = 0, 



by the transformation 



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69.] HOMOGENEOUS FORMS 203 

and therefore 

that is, 

F{vii, Wj, M'i) = 0, 

on rejecting the i'actor e'"/?*', which occurs because F is a ternary 
form (say) of order in. Hence it follows that when tlie integrals of 
a liiiear equation of the third order are algebraic Junctions, a 
homogeneous relation of finite order exists among any three linearly 
independent integrals. 

Moreover, when any other set of fundamental integrals F,, Y,, 
Y, is taken, we know that 

y, = a,Y, + a,Y., + a^Y, 
y^=h,Y, + b,Y, + b,Y, 

where the coefficients a, b, c are constants. The variables in the 
homogeneous ternary form are therefore subject to linear trans- 
formation; and thus the theory of ternariants can be associated 
with those homogeneous linear equations of the third order, which 
have tbeii' integrals algebraic. The various cases will arise 
according to the order of the form F; this order is always 
greater than unity, because the integrals considered aie linearly 
independent. 

If, still further, we choose to combine the geometry of the 
ternary form with the form in its association with the equation, 
then the preceding algebraic relation ^ = is the equation of an 
algebraic plane curve referred to homogeneous coordinates r the 
curve is usually called the integral curve. 

equation of the fourth 



Wo 
order 


may 


proceed similarly with a 


n ec 








dw 



when all its integrals are algebraic. If we choose, we may trans- 
form it by the relation 



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204 EQUATIONS OF THE 

the quantity e^^^'^ must be aigebraic, because 



where (7 is a non -vanishing constant ; and the equation in y, 
which is of the form 

dz' dz^ as ■' 

has all its integrals algebraic. Taking any four linearly inde- 
pendent solutions y-i, y^, 1/3, y,, and writing 

then as p, a-, t are algebraic functions of z, they must be given 
by three equations of the form 

or of simpler equivalent forms, which are completely algebraic in 
character. Eliminating z between the first and second, and also 
between the first and third, and taking the eliminants in a form 
free from irrational quantities if these occur, we have two 
equations 

F,{p,,,T)=0. ff.(p.^,T)=0, 

two non- homogeneous polynomials in p, a, r. Replacing these 
quantities by their values in terms of ^1,^2, 1/3, 2/4, and multiplying 
ea«h equation by the power of 1/,, appropriate to free it from 
fractions, we find 

where F and G are homogeneous polynomials in their arguments 
or, in other phrase, are quaternary forms in 7,, y^, y^, ya- As in 
the case of the cubic, these equations imply the fiirther equations 

F(Wj, Wa, Wj, W4) = 0I 

so that, when the integrals of a homogeneous linear equation oj the 
fourth order are algehraic Junctions, two homogeneous relations of 
finite order earist among any four linearly independent integrals. 



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69.] FOURTH ORDElt 205 

Again, when the variables i/j, y-s, y,, y^ are replaced by any 
other set of fundamental integrals F,, Y^, Y^, Fj, the two sets of 
variables are connected by homogeneous linear relations; and 
thus the theory of quaternariants can be associated with those 
homogeneous linear equations of the fourth order which have 
their integrals algebraic. The various cases will arise according 
to the orders of the forma F and G; these orders are always 
greater than unity, because the integrals y,, y^, y^, y^ are linearly 
independent. 

We may also combine the geometry of quaternary forms with 
the forms themselves as associated with the equation. In that case, 
each of the equations F= 0, G = is the equation of a non-planar 
surface in three dimensions referred to homogeneous coordinates : 
the two equations combined determine a skew curve, which ac- 
cordingly is the integral curve. 

Similarly, in the case of equations of the fifth order, of which 
all the integrals are algebraic, we have three homogeneous non- 
linear relations among any fundamental set of integrals ; and there 
are corresponding associations with the theory of homogeneous 
forms in five variables and the allied geometry. And so also for 
linear equations of higher orders. 

Note 1. There cannot be two homogeneous relations among a 
set of three linearly independent integrals of an equation of the 
third order: for they would determine a limited number of sets of 
constant values for the ratios y, : y^: y^, contrary to the postulate 
of linear independence. 

Similarly, there cannot be three homogeneous relations among 
a set of four linearly independent integrals of an equation of the 
fourth order; for their existence would imply a corresponding 
contradiction of the same postulate. And so for other equations 
of higher ordei-s. 

It might however happen that, for an equation of the fourth 
order, only a single homogeneous relation exists among four 
linearly independent integrals; that, for an equation of the fifth 
order, the number of homogeneous relations among a fundamental 
set of integrals is less than three ; and so on. If the relations thus 
given in each of the respective cases are the maximum number of 
homogeneous relations that can exist, we can infer that not all 



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206 BINARY [69. 

the integrals of the respective equations are algebraic: and a 
question arises as to the significance of the respective relations. 

iVofe 2. The converse of the general argument must not be 
assumed valid : that is to say, the existence of a homogeneous rela- 
tion between the members of a fundamental system of integrals 
of an equation of the third order is not sufficient to ensure the 
property that all the integrals are algebraic. Thus we know 
that a number of transcendental functions of a variable can be 
connected by algebraic relations : and such instances are not the 
only possible exceptions. 

70. The preceding method of a.ssociating the theory of forms 
with linear equations does not apply directly when the equation 
is of the second order : for a homogeneous relation between two 
integrals would imply one or other of a limited number of con- 
stant values for the ratio of the integrals, which accordingly 
could not be linearly independent. This deficiency, however, is 
rendered relatively unimportant, because Klein's method explained 
in §1 59^62 for the equation of the second order gives the 
complete solution of the question propounded as to the cases 
when ail its integrals are algebraic. The results there given 
can be (and have been) obtained by processes directly connected 
with the theory of binary forms. After the preceding exposition, 
the analysis is mainly of formal interest, and adds little to 
the knowledge of the solutions regarded as functions of the 
independent variable. 

It will be sufficiently illustrated* by one or two examples. 

Ex. I. We tako the differential equation in the foi'm 

and consider the value of a homogeneous polynomial function of two integrals 
^1 aud y^i linearly independent of one another. Let this polynomial be of 
order re, and write 

* For fuller diBOUsaion and details, see Faaks, Crelle, t. Lixii (1376), pp. 97— 
142, f6.,t. Lxxxv (1878), pp. 1—25; Briosclii, itfalft, ^nn., t. zi (1877), pp. 401— 411; 
Forsyth, Quart. Journ., t. ixm (1889), pp. 45—78. 

A memoir by Pepin, "Methode pour obtcnirlesintfigralesalgSbriques des Equations 
dififeentialles lin^aires du seeond ordre," Row. Ace. P. d. N. L., I. isxiv (1883), 
pp. 243 — 389, may slso be consnlted with advantage. 



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70.] FORMS 207 

saj, where s is the quotient ^,-^-^3. When substitutioa is made for i/j and 
^2 ill terms of x, let the vahie of/ be <p (x), so that 

Now \i R{yi,y^ = H{f) be the Hessian of/, and if H{u) ho the Hessian 
of a, so that 

^W"-<-'>-(»S-<»-')(S)'}- 



We have also 








*! 


*■ 




■"&;- 


y,SJ-eon.i 


say, so that 




-■£=^ 


Now 




j,-«-*Wi 










• *, 


, c 1 <;» 




^S <^ 



Differentiating, and aubatituting for the second derivative oiy^, we have 

y^\dx J j/ dx u ds y^ d^ ds^ 

Multiply by n, and add the squaj^s of the aides of the preceding equation : 
^n^21 L d^logu) \_ fdu\^\ _ I /d<p\\ d^{los<i>) 

The coefficient of CVa"' on the left-hand side is 

60 that 

the Heaaian in terms of functions of x : let this be written 



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208 EXAMPLES [70. 

If now * (y^, y^) denote the oubiuo variant of/, so that 
1 rofdH of a//) 

then, proceeding in a similar way, we find 

And so for other covariants. 

As a special case*, let it be required to find tho value of 0, if when the 
binary form is tho quadratic 

a„ylH2a^3/lya + a32'2^ 
c^ {x) is a root of some rational function of x. In this instance, 

a constant ; hetice i^ {:c) is either a rational function, or is the square root of 
a rational function. The integration is immediate ; for 

d$ Cdx 



a^s^ + ^a^s + a^ if>ixy 
The value of s is thus known : and the consequent values of y^ and y^ a 
immediately given +. 

.Ec. 2. Shew that, if the integrals of the equation 



and is a root of some rational function of x, then <^* must be rational ; and 
obtain tho relation between / and if) (x). 

Ex, 3. The integrals of the equation 



and (a;) is a root of some rational function of x ; shew that, unless ^ {x) is 
actually rational, the quadrinvariant of the binary quartic must vanish. In 
either case, find the relation between I and 1^ {x). (Brioschi.) 

* Fuchs, CreiU, t. lxxsi (1876), p. 116. 

t See my Treatise on Differential Equations, % 62. 



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70.] OF BINARY FORMS 

Ex. 4. Find the value of / in the equation 

when, in the relatioii 

connecting two integrals, the function ^ is supposed known. 
Ex. f). Shew that, if two integrals of the equation 

are connected by a relation 

where A, B, C, D are cojistaiits, then 

Assuming the condition satisfied, integrate the equation. 
Ex. 6. Two integrals of the equation 






are connected by a relation of the form 

Ay^^^By^h,^ + Gy,y^^+Dyi^E=0, 
where A , B, G, D, E are constants : prove that 



d^Q .„dQ 



-z(P-^Gl^q=Q. 



Shew that the quantity on the left-hand side of this conditional equation ia 
invaiiantive for change of the independent variable ; and hence, assuming 
the condition satisfied, shew that the equation can be transformed so as to 
become a particular case of Lamp's equation (Chap. ix). (Appell.) 



Equations of the Third Order and Ternariants. 

71. Returning now to the differential equation of the third 
order in the form 



and supposing that all its integrals are algebraic, we proceed to 
consider the equation 



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210 EQUATIONS OF THE THIRD ORDER [71. 

where F is a homogeneous polynomial in any three linearly 
independent integrals. For this purpose, it will be convenient 
to have an equivalent simpler form of the equation which is given 
by a known transformation*, viz, we have 





a;+^-»' 


dt 


■ 'iih^-if 


we take 


I-- 



the last of these relations may be replaced by the equation 

The equation among any three integrals is 

Consider the simplest case ; it arises when «. = 2, so that F is 
then a quadratic polynomial involving six terms. Writing 

a« = a,u, + «iMa + (laMa, 
where a„ a^, a^ are umbral symbols, the equation can be symbolic- 
ally represented by 

We have 

where u' is du/dt, and so for u". Differentiating again, and 
replacing u'" by — lu, we have 

that is, 

<*„'««" = 0, 
on using the original equation. Similarly, on differentiating this 
result, 

- la^aa- + Ou"^ = 0, 
that is, 

«„'■= = 0, 

" See a paper by tlie author, Fbil. Tram., (1888), p. 441. 



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71.] AND TERNARIANTS 211 

Oil using the first derivative of the original equation. Differen- 
tiating once more, we have 

la^au" = 0, 
80 that either / = or a^Ou" = 0, 

If / is not zero, then we must have 

and therefore, by the second derivative of the original equation, 

au^ = 0. 
Hence, on the present liypothesis, we have 
a^^ — 0, aaa„' — 0, a„^=0, auOu" = 0, au'aH" = 0, oi„"' = 0. 

Now each of these equations is linear and homogeneous in the 
six real coefficients that occur in a^^; eliminating these coeffi- 
cients, we obtain, as equal to zero, a determinant which is the 
fourth power of 



M2 . ■ 

and the latter ought therefore to vanish. But because w,, u^, Ms 
are linearly independent, this determinant (being the determinant 
of a fundamental system) dues not vanish — it is a non-zero 
constant in the present case. Accordingly, the hypothesis that 
/ ie not zero is invalid. 

Hence i" = ; and therefore, on returning to the original 
equation, we have 

Writing 

our original equation becomes 

dz^ dz dz ^ 

Any three linearly independent integrals are connected by a 
quadratic relation 



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212 TEENABIANTS [71. 

To obtain the integrals, we note that one value of m is a constant, 
say unity ; thus 

where 

Thus three integrals of the original equation are ^i^, O^O^, 8^, 
where fi and d^ are two linearly independent integrals of the 
iatter equation of the second order. 

It may be noted that three independent integrals of the 
M-equation ai'e 1, (, f\ so that 

dt ^ dt , dt 

I'd,-^- "'S-'' I'di"'- 

and therefore 

y^i - Vi'h = 0' 
thus verifying the existence of the (quadratic relation obtained in 
a canonical form, 

Assuming known, we have 

dz~ 6^' 
so that 

and thus three integrals of the original equation are 

e-. e.l%. ..|/|f. 

The comparison of these integrals with 6^, Oid^, 6j' is immediate ; 
for it is a well-known theorem that, if ^i is a solution of an 
equation 

g + pe.o, 

then another solution, which is linearly independent of Sj , is given 

by 

Denoting this by 0^, the above three integrals are at once seen to 
be ft", eA, «.'. 



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71,] EXAMPLES 213 

Ex. 1. Prove tha,t, if a be a solution of the equatioD 

the primitive can be expressed in the form 

i/ = Au + Binis.i){a I — J+(7«espf — a |— J, 

wbere A, B, C are arbitrary constants, and a ia a determiuate constant. 
What 13 the primitive when a vanishes ? (Math. Trip. Part i, 1895.) 

Ex. 2. Prove that, if three linearly independent integrals of the equation 

be connected by a relation F{;y^, y^, 3'3) = 0, where ii™ is a homogeneouiS 
polynomial of the third degree, then / muist satisfy the equation 
(567'= - 48/7") 7"'+5477"'«- 1447'/"i'"+ IS^ . 7737'" +^ 24S7'3 
- 7 . ZQ^Prr + 84"77'S + ^^^^ /* = 0. 

Ex. 3. Prove that, if both the fundamental invariants* of an equation of 
the fourth order vanish, so that it can be taken in the form 
y -hlOPV +l^P'y +{'il' +97«)./ = 0, 
then fDur 1 neirlj mdejiendent integrals are given by 8-^, 6-^6.^, 6i6.^\ 6^, 
where S^ ml fl ire Imeiily indpi>endent intetjrali tf 

Shew also that, if the relations 

* These arise in the aame mannec as for the oubio. It the etiuation 
be transformed by the relations 



"4 + 40„p + Q,« = 0, 



sS-^^-=0^ 



and the fundamental invariants are Os' ^»~^(ii'' ^^ "^ ' 
p. 210, note. 



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214 EQUATIONS OF THE THIRD ORDER [71. 

sulsist among four linearly independent integrals of aa equation of the 
fourth, order, (so that the integral curve is a twisted cubic), tho equation 
must he of the above form. 

Ea:. 4. Construct the equation of the fourth order having fli^i, 5i0ai 
6^1, ^2^3 for a set of linearly independent integrals, whore 6^ and fl^, -p^ and 
^2, are linearly independent int^rals of the respective equations 






s+™-". sa+e*-"- 



Hence infer the form of a quartio equation when a single homogenec 
quadratic relation subsists among a fundamental system of integrals. 

Ex. 5. Shew that the equation 

y"' + ry"+4^' + (6s' + 4r.s)y + 3(s"+!-s')j=0 
is satisfied by ;/ = ^, where 8 is an integral of 
6"+sB=0; 
and hence integrate the equation. (Fan 

Es. 6. Shew that, if five linearly independent integrals of an equation 
the fifth order are connected by the relations 

[1 ^1. Vi, Vs, 3'* ||=0. 
11 ^2. ys> ^'4. ^i I! 
the equation can be taken in the form 



Sj-ao-Sj-iioiSft 






.:;e-('»£-'-')i-(*S+«'£)»-»^ 



and thence integrate the equation as far as posisible. (Fano.) 

72. Consider now the more general case when three linearly 
independent integrals of the equation 

are connected hy an iiTesoluble relation 

*■&.,</„ y.) = o, 

where J*' is a homogeneous polynomial of order greater than two : 
the question is as to the character of the integrals of the equation. 
For the discussion, it is assumed that the differential equation 
has its integrals regular and fi'ee from logarithms: it thus is of 
Fuchsia n type. 

Let K denote any non- evanescent covariant of the quantic F; 
such a covariant is the Hessian, which would vanish only if F 



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72.] AND TEUNARIANTS 215 

contained a linear factor. Let z describe any contour, which 
encloses any one of the singularities, and return to its initial 
value; the effect upon the fundamental system of integrals iji, y^, 
yn is to change them into another fundamental system Fi, F,, F3, 
the two systems being connected by relations 

Y,= a,y, + ^,y, + rf,y„ (r=l, 2, 3). 

The determinant of the coefficients a, /3, 7 (say A) is different 
from zero in every such case ; in the present case, owing to the 

absence of the term in A^ from the equation, we have {§ 14) 

A = l, 
by Poin care's theorem. 

Now the preceding relations constitute a linear transformation 
of the variables in the foregoing homogeneous forms ; hence if ^ 
be the index of K, and ^denote the same function of Fi, Fj, F; 
as .ff" is of y,, j/g, j/j, we have 

= K, 

for fi is necessarily an integer. It thus appears that the value of 
K is unaltered by the description of the contour. 

This holds for each of the singularities, as well as for s = qo ; 
hence K, when expressed as a function of z, is a uniform function. 
To obtain the form of K in the vicinity of any singularity a, we 
take account of the fact that the equation is of Fuchsian type : 
hence in the vicinity we have, for any integral y, 

(e — a)~py = hoiomorphic function oi z — a, 

where |^| is a finite quantity. Now K is of finite order in the 
variables 1/1, y,, y,; accordingly substituting for them, and remem- 
bering that ^ is a unifoi'ni function of 2, we have 

{z — a)~''K = hoiomorphic function of « — tt, 

where <r is an integer, positive or negative. This holds for estch of 
the singularities, the number of which is limited when Q and R 
are rational functions of 2 ; it holds also for z=ci:> . Hence K is 
not merely a uniform function, but it is a rational function, oi z. 



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216 APPLICATION OF [72. 

It therefore follows that every eovariant of the quantic F is 
a rational function of z, exceptions of course arising in the case 
when the eovariant in an invariant, so that it is a mere constant. 

Take then any two covariants, say the Hessian H, and any 
other, say K : we have 

where and ijf Eire rational functions of z. These are three 
algebraical equations to determine y,, y^, y^ in terms of z; and 
therefore the differential equation is integrable algebraically, a 
theorem first announced* by Fuchs. 

A case of exception arises, when the Hessian is a constant : the 
quantic F is then of the second order so that the case has already 
been discussed ; the integration of the original equation depends 
upon the integrals of a linear equation of the second order. 

As an illustration, consider the equation 

wheii a fundamental set of integrals is coanected by a homogeneous cubic 
relation. We assume that the equation is of Fuchsian type. 
Talcing the cubic in the canonical form, we have 

I being a constant. The Hessian is a rational function, say 1^(1+8?^); so 

ir=f (y.'+y,' +y,=) - (1 + 2z=) y,y,j<3 = <j. { 1 + 8;«), 

and therefore 

Taking the other symmetric covariantt of the cubic, which also ia a rational 
function, we have 

and * is equal to a rational function ; so that, ta.kiiig account of the above 
value of ^I'+ya^+^s', we can write 

Thus jijS, ^^^, yj' arc the roots of 

' Acta Math., t. I (1882), p, 830. 

t Cayley, Coll. Math. Papers, t. si, p. 345, 



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72.] COVABUNTS 217 

an irreducible cubic. So far aa the coefficients are concerned, they are known 
to be rational functions of z ; the denominator of each such function is known, 
because its faotora arise through the aingularitiee of the equation and the 
multiplicity of any factor can be determined through the associated iadiciaJ 
equation ; and the degree of the numerator has an upper limit, determined 
by the behaviour of the integrals for large values of z. Heace and ■^ can 
be regarded aa known, save aa to a polynomial numerator in each case. 

We have 



Tl"' = Afi^ + Bir, + a^ ] 

the last three being obtained, after diftereatiation, by repeated use of the 
cubic equation for ij, and the quantities A, B, G, ... being functions of </i, i/' 
and their derivatives Now writing y = i^ in the differential equation, we 
find 

When the above values are substituted and the result is reduced by means of 
the cubic equation, so that no power of ij higher than the second occurs, we 
have an equation of the form 

where Y,, Yj, Y, involve 0, ijr and their derivatives, and are linear in Q, R. 
As the cubic is irreducible, so that this equation holds for each root, we have 

Yi = 0, Ya^O, Y3 = 0, 

three equations to determine and ■^. There consequently exists a relation 
among the remaining quantities, viz. Q and R : and this must be equivalent 
to the condition (§ 71, Ex. 2), which must be satisfied in order that the 
equation .^=0 may exist. 

Similar results hold for the cubic equation, when the homo- 
geneous relation between the integrals is of order greater than 
three ; and corresponding results hold for linear differential 
equations of higher orders. In fact, if a general homogeneous 
relation of finite order higher than the second subsists among a 
fundamental si/stem of integrals of a linear differential equation of 
order n, then the equation is integrable algebraically: the proof 
follows the lines of the preceding proof exactly. 

This range of investigations will not, however, be pursued 
further, as it becomes mainly formal in character, depending upon 



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218 APPLICATION OF COVARIANTS [72. 

the theory of covariants and upon the application of the theory of 
groups to linear differential equations. An excellent account of 
what has been achieved, together with many references, is given 
in a memoir* by Pano who has made many contributions to tho 
subject ; a memoir-f- by Brioschi contains some investigations con- 
nected with ternariants; and other detailed references are given 
in Schlesinger's treatise^, which contains ati ample discussion of 
the subject. 

' Math. Ann., t. Liri (1900), pp. 493—590. 
+ Ann. di Mat., 2" Ser., t. xm (1885), pp. 1~21. 

J Tkeorie der linearen DiffeTentiatgUiekimgen, ii, 1 (1897), pp. viii — si. The 
diBCUssioti is to lie found in ohupters 2 — 6 of the tenth seotlon of the treatise. 



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CHAPTER VI. 

Equations having only some of their Integrals regular 
NEAR A Singularity. 

73. It has been seen that, if all the integrals of an equation 
are to be regular in the vicinity of each singularity, the coefficients 
in the equation must be rational functions of z of appropriate 
form and degree. 

It may, however, happen that the coefficients are rational 
functions of s but are not of the appropriate form and degree : 
in that case, it is not the fact that all the integrals are regular, 
and it may even be the fact that none of the integrals are regular. 
This deviation from regularity need not occur at each singularity 
of the equation : a fundamental system may be entirely regular in 
the vicinity of one (or more than one) of the singularities, and 
may not possess its entirely regular character in the vicinity of 
some other. The conditions necessary and sufficient to secure 
that all the integrals are regular in the vicinity of a singularity a 
have already (Ch. Hi) been obtained. If these conditions are not 
satisfied, then the composition of the fundamental system in the 
vicinity of the singularity a is no longer of an entirely regular 
character; we desii'e to know the deviations from regularity. 

It may also happen that not all the coefficients are rational 
functions of z; in that case, if uniform, they are transcendental 
functions and possess at least one essential singularity, say c. 
Further, owing either to a possibly excessive degree of the 
numerator in a rational meromorpbie coefficient or to a possibility 
that z—<x: is an essential singularity of some one or more of the 
coefficients, it can happen that the conditions for regularity of 
integrals near a-^oc are not satisfied. The fundamental system 



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220 EQUATIONS HAVING [73. 

is then not entirely regular near c or for large values of \z\, in 
the respective cases indicated, and it may even be devoid of any 
regular element; the same question as to its composition arises 
as in the corresponding hypothesis for the singularity a. 

Accordingly, for our present purpose we assume that the 
coetBcients in the differential equation are everywhere uniform: 
that (unless as otherwise stated) they may have any number of 
poles, and that they may have one or more essential singularities. 
When tt is a pole of one (or more than one) of the coefficients, 
and is not an essential singularity of any of them, we have one 
of the cases just indicated; when qo is a pole of coefficients, 
not being an essential singularity of any one of them, we have 
another. We write 

1 



in these respective cases ; and then our differential equation takes 
the form 

d^w d^-'w d'^'^w dvj 



where the point a; = is a pole of some (and it may be of all) the 
coefficients. If a!I the integrals were regular in the vicinity of 
ic = 0, then x'^p^ for r = 1, 2, ..., m would be a uniform function of 
X that does not become infinite when x=0. As some of the 
integrals are to be not regular in the vicinity of x = Q, the 
multiplicity of the origin as a pole of p, must be greater than r, 
for some value or values of r. Let 

p^=a:-'^'Pr{s>), (r=l, ..., m), 

where m-, is a positive integer (which may be zero for particular 
coefficients), and Pr (*') is a uniform function of a: which does not 
become infinite when a:=0: also it will be assumed that, unless 
pr vanishes identically, ■=r, has been chosen so that Pr{0) does not 
vanish, so that ct, measures the multiplicity of the pole of p, at 
the origin. Then one or more than one of the quantities 

w,.-r (r = l,...,m) 

is a positive integer greater than zero. 
As in § 23, let 

r 1 ,,.... 



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73.] ONLY SOME INTEGRALS REGULAR 221 

and suppose that 

■^lA + 0A-li + 0A-5i>+ ... + 0it*~' + 0o£* 

is ao integral of the equation, regular ill the vicinity of a; = and 
belonging to an exponent fj. ; then it is known (§§ 25 — 28) that 
<^„ is a regular integral also belonging to the exponent fs,, so that 

where <!>„ is a uniform function of x which does not vanish when 
x = 0. As this expression, when substituted for w, should make 
the equation satisfied identically, the aggregate coefHcient of the 
lowest power of x must vanish (as, of course, must all the other 
aggregate coefficients). The lowest power of ic in the respective 
terms has for its index 
/j.~m, ij. — ^i—{m — i), fi. — 'a^ — {'in — 2), ..., /j. — ia^^i — 1, /j. — 'nr^: 

and for any other integral, belonging to an exponent <7, the 
corresponding numbers would be 

a — m., (r-i!Ti — (m ~1), 17 ~ t^s — (m — 2), ..., <r — OTm_i— 1, <r — st™. 

Let 

w. + (m-s)=rj„ (s = 0, 1, ...,'m), 

and consider the set of integers 

n„, n., .... n^. 

Of these, let the greatest be chosen. It may occur several times 
in the set ; when this is the case, let the first occurrence be at 
n„, as we pass in the order of increasing subscripts, so that 
n^<n„ , for r = 0, 1, ...,n-\, 
n„^n^., r = 0, 1, ....m-n. 

Then n is called* the characteristic index of the equation : when 
K = 0, all the integrals are regular. 

The lowest power of x after substitution of the expression for 
the regular integral has (U. — n„ for its index ; it arises through 

p„ — y- — and later terms in the differential equation ; as the 

coefficient of this- lowest power must vanish, the exponent fi must 

* Thome, Ci-elle, t. lksv (1873), p. 267. 



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222 CHARACTERISTIC INDEX AND INDICIAL EQUATION [73. 

satisfy an algebraic equation of degree m — n. Similarly for an 
exponent o- to which any other regular ititegi'al belongs ; it also is 
a root of the same algebraic equation; and each such exponent 
satisfies that same algebraic equation of degree m — n, which 
accordingly ia called the indicial equation. But it must not be 
assumed (and, in fact, it is not necessarily the ease when n > 0) 
that the number of regular integrals is equal to the degree of the 
indicial equation. It is clear that, in all cases where n.>0, the 
degree of the indicial equation is less than in. 

71. Suppose now that the given differential equation of order 
■m has a number s of regular integrals, which are linearly inde- 
pendent of one another, where s <m: (the case s = m has already 
been discussed) : and that there do not exist more than s linearly 
independent integi-als. After the earlier discussion of fundamental 
systems, it is clear that any regular integral of the equation is 
expressible as a homogeneous linear combination of the s integrals, 
with constant coefficients ; also that, if every regular integral of 
the equation is expressible as such a combination of s (and not 
fewer than s) such integrals, the number of regular integrals 
linearly independent of one another is s. 

Further, a linear relation among the integrals of the equation, 
involving a number of regular integrals and only a single one that 
is not of the regular type, cannot exist ; for the single non-regular 
integral would involve an unlimited number of negative powers of 
w, while each of the others occurring in the linear relation involves 
only a limited number of such negative powers. 

A linear relation might exist among the integrals of the 
equation, involving a number of regular integrals and two Integrals 
that are not of the regular type. We then regard the relation as 
shewing that the deviation from regularity is the same for the 
two integrals : and in constituting the fundamental system for the 
equation, we could use the relation as enabling us to reject one 
of the non-regular integrals, because it is linearly expressible in 
terms of integrals already retained. So also for a linear relation 
with constant coefficients between regular integrals and more than 
two integrals of a non-regular type. 

Again, suppose that our differential equation of order m has 
an aggregate of n integrals, regular in the vicinity of ic = and 



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74.] THEORY OF BEDUOIBILITY 223 

linearly iniiependent of one another; and let it be formed of sub- 
groups of integrals of the type 

for'K — 0,l,2,...,ic, where 



f-A.■v^^,lZ'■-' + ^^l,,i^ 



Then, after ^ 25 — 28, we know that these n linearly independent 
integrals constitute a fundamental system for a linear differential 
equation of order n, the coefficients of which are functions of ic, 
uniform in the vicinity of a^ = ; let it bo 









r^^^ 
'^'d^'^ 



+ i'j(^ =0- 



Now this equation, being of order n, cannot have more than n 
linearly independent integrals i and its fundamental system in the 
vicinity of gs — O is composed of the n regular integrals of the 
original equation. Hence, by § 31, we must have 

r^ = a>-''B^(w), {fi = l, 2, ..., n), 

where Il^{ic) is a holoraorphic function of x in the vicinity of 
ic = 0, such that R^(0) is not infinite. Accordingly, the aggregate 
of the n linearly independent regular integrals of the original 
equation are the n integrals in a fundamental system, of a linear 
equation of order n of the foregoing type. 



Eeducibility Of Equationk. 

75. If therefore some (but not all) of the integrals of the 
given equation of order m are of the regular type, it has integrals 
in common with an equation of lower order. On the analogy of 
rational algebraic equations, which possess roots satisfying an 
algebraic equation of the same rational fonn and of lower 
degree, the differential equation is said to be reducible. 

Consider two equations 



) d™ '■y 



*(!/)-«. 



*Sj 






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224 BEDUOIBILITY OF [75. 

■where m>n; and take an expression 

where the coefficients fio, .B,, ..., Ri are at our disposal, and 

Let these disposable coefficients bo chosen, so as to make the order 
of the equation 

JJ(y)-iliT(!,)! = 

as low as possible. By taking the ^ + 1 relations 

P. = B.Q,. 

-p,=-R,a+-B.(ia'+a), 

P, = ftft +B,{(i-i) q; + Q,] + B. lii (i - 1) Q." + iq; + a), 

i'i-ae.+-Bi-.(a'+ft)+-Ki-.(a"+2Q.'+Q.)+..., 

which determine R^, ■..,Ri, we can secure that the terms involving 
derivatives of y of order higher than n — \ disappear. Accordingly, 
writing 

where S^, S,, ..., Sj are determinate quantities and 

we have 

where K is of order less than if. Moreover, if P„, ..., P^, 
Qo, ■■■, Qn are uniform functions of x, having x = either an 
ordinary point or only a pole, the same holds of the coefficients R 
and the coefficients S ; so that L and K are of the same generic 
character as M and N. 

From this result several conclusions can be drawn, 
I. Any integral, common to the equations Jf = 0, N = 0, is an 
integral of the equation K = 0. If, therefore, every integral of 
JV = is also an integral of M=0, it follows that K=0 must 
possess n linearly independent integrals ; as its order is less than 
n, the equation is evanescent, and we then have 
«{s).ifiV(S,)). 



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75.] AN EQUATION 225 

II. Any integral, common to the equations JV=0, K~0, is 
an integral of the equation Jf = ; and therefore, in connection 
with the first part of the preceding result, the integrals common 
to M^O, N = constitute the integrals common to i^'" = 0, K=0. 

The process of obtaining the integrals (if any), common to 

two given equatioDS ilf =0 and i\r = 0, can thus be made a kind of 

generalisation of the process of obtaining the greatest common 

measure of two given poiynomiais. Proceeding as above, we have 

M =iiV +K \ 

K = L,K, +K,r 

where K^, K.2, ..., Kg are of successively decreasing orders. Then 
unless an evanescent quantity K of non-zero order is reached, 
sooner or later a quantity K is reached which is of order zero, 
that is, contains no derivative. 

In the former case, let ^,+1 be evanescent ; then the integrals 
of the equation Kt= constitute the aggregate of integrals common 
to M = 0,N=0. 

In the latter case, let Kg be the quantity of order zero ; then 
the integrals common to M=0, ^ — are integrals of 

lf, = y/W-0. 
Now f(z) is not zero, for otherwise ^s would be evanescent ; and 
therefore we have 

y = 0, 

the trivial solution common to all homogeneous linear equations. 
We then say that Jf = 0, A'" = have no common integral. 

III. An equation having regular integrals is reducible. For 
one such integral exists in the form 

y— :•/("). 

where |^| is finite, andy(fl;) is holomorphic in the vicinity of «= 0, 
while /(O) ia not zero. We have 

ydw X f{x) 



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226 CHARACTERISTIC FUNCTION [75, 

where R(x) is a holomorphic function in t.he vicinity of w = 0, 
such that R{0) is not zero. Thus the given differential C(^iiation 
has an integral satisfying the equation 

that is, it has an integral common with an equation, which is of 
the first order and is of the same form as itself: in other words, 
the equation is reducible. 

But it is not to be inferred that such equations are the only 
reducible equations. 

IV. If an equation M = has p (and not more than p) 
linearly independent regular integrals, it can be expressed in 
the form 

where N is of order p, and L is of order m — p. 

For the p regular integrals are known (§§ 25—28, 74) to 
satisfy an equation of the form 

of order p. Every integral of iV"=0 is an integral of M=0; 
whence, by I., the result follows. 

76. We proceed to utilise the last result in order to obtain 
some conclusions as regards the regular integrals (if any) of a 
given equation, say, 

,^ , , d'"w! d'^^'w dw 

The result of substituting x^ for w in P (w), where p is a constant 
quantity, is 



this is called* the characteristic /unction of the equation 7' = or 
of the operator P. We have 

... +p,^-,^+Pv,; 

* Frobeaius, Crelle, t. i.ssx (lS7a), p. 318, 



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76.] INDICIAL EQUATION 227 

when the right-hand side is expanded in ascending powers of x, it 
contains (owing to the form of the coefficients p) only a limited 
number of powers with negative indices. The highest powers of 
oT^, arising out of the m + 1 terms in i>;~''P(W), have exponents 
m, OT,+m-J, OT2 + m-2, ..., ct™_, + 1, vr^, 

Ho, n., ..., n^. 

Let n be the characteristic index of the equation, so that n„ is 
the greatest integer in the set : if several of the quantities 11 be 
equal to this greatest integer, then n^ is the first that occurs as 
we proceed through the set from left to right. Denoting the 
value of n„ by g, let 

''^"ffl-^"?'* (*')"?'•' ^'" ~ ^' "' ■■■' ^)' 

so that 5„ (0) is not zero, and no one of the quantities q^ (0) is 
infinite. Then 

«-.P(i')_^-.G(p.«;), 

where (? is a polynomial in p and is hoiomorphic in a: in the 
vicinity of a; = 0. Moreover, expanding G(p, x) in ascending 
powers of x, we have 

Gip,^)=9,{p)+a:grip) + ..., 
where each of the coefficients ^ is a polynomial in p, of degree not 
higher than m; the degree of ffoip) is m — n, and the degree of 
gg^mip) is ™- Also, g^ip) is the quantity called (§ 39) the indicial 
function ; the equation 

ft(f>) = 
is called the indicial equation. 
Now take 

N(w)^a:ffP(w) 

d^w ^ , d'"-"''!!! dw 

- 1-^ -£~ + *' £?-T + ■ ■ ■ + «-'^ S + *-'"• 

where q„ = xs~'^ ; the equation P — can manifestly be replaced 
by the equivalent 

J{»)-0, 

which is taken to be the normal form for the present purpose. 
We have 

arPN{af) = G{p,x) = g,{p)-\-xg,{.p) + --, 



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228 NORMAL FORM OF EQUATION [76. 

which thus contains only positive powers of x when the equation 
is in its normal form, and which has the indicial function for the 
term independent of x. 

We have seen that, if P {w) = possess regular integrals, it 
is a reducible equation : and the operator P can then be repre- 
sented as a product of operators. Consider, more generally in the 
first instance, two operators A and B, each in its normal form ; and 
let G, also an operator, denote AB. Further, let the characteristic 
functions of A, B, G, respectively be 

A (^) = a^^fia:, /,) = «^ 2 /. (p) ^^ = 2 /, (p) a:^+- 

B{a^) = a^g(x,p)=^x<' 2 g^ip)x'^= 2 g^{p)x'^+>- 
0(a:'') = xi'h{x,p)^af 2 h^(p)x''== 2 k^(p}x^+<-, 
where the summations in f(x, p) and g{x, p) include no negative 
powers of fl^, because A and B are in their normal forms. Now, as 
C= AB, we have 

G{x'-) = AB{af) 

^A[%j,{p)ar+'-] 

= %^l^gAp)/>.(^+p)x^+'^+^ 
and therefore 

S 4, (p) «' - ^X ^S J7, (p)/, (M + p) «="'. 

As X and p. are incapable of negative values, there are no negative 
values for a- ; and therefore G is in a normal farm. Also 

fh(p) = gi,{p)f<,{p), 
80 that the indicial function of G is the product of the indicial 
functions of its component operators : and 

h,(p)= S^y„ (/>)/„_„ (/i + p). 

Further, if C be known to possess a component factor B which, 
when operated upon by A, produces C, then A can be obtained. 
For, take B and C in their normal forms : the equation 
S h,(p)i^~ t Xsr,(p)A(^ + p)a^+. 



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76.] CHAEACTERISTIC INDEX 229 

then holds. The values of X are clearly 0, 1, ..., so that A is then 
in its normal form ; and the successive quantities fy are given by 
the equation 

for o- = 0, 1, ..., p, the values obtained being polynomials in p, 
because G is known to be composite of A and B. 

Of course, this merely gives the characteristic function of the 
operator ; but the characteristic function uniquely determines the 
operator. For let /(ic, p) be a function, which is a polynomial 
in p, and the coefficients of which are functions of x: and let 
the degree of the' polynomial be m. Then we have* 

where, taking finite differences in tlie form 

4/(«.p) =/('»,(' + !)-/(«,/>), 
we have 

<.!«. = 14-/(«,P)1.... 
Tlius 

which is the characteristic function of the operator 



da?'* "* ' dai"-' 



. + v^x -J- + u„: 



the operator is determined by the characteristic function. 



Characteristic Index, and Number of Regular Integrals. 

77. Now let the equation of order m, taken in its normal 

form, be 

„, , d^""w ,d"'-hjj dw „ 

i\r (^) . ^^- _- + 5,.-. ^^^ + . .. + 5™-,«^ ^ + 3™«. = ; 

and suppose that it possesses s (and not more than s) regular 
integrals, linearly independent of one another. These s integrals 



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230 CHARACTERISTIC INDEX AND [77. 

are a fundamental ayatem of an equation, of order s and of Fuchs- 
ian type ; when this equation is taken in its normal form, let it be 

^ ' d^ oaf-' da> 

where cr,, ctsi -■■. <^a ^^^ holomorphic functions of x in the vicinity 
of x = 0. As all the integrals of jS' = are possessed by iV=0, 
there exists a differential operator T of order m — s, such that 

because N and S are in their normal forms, T also is in its normal 
form, so that we can take 

(M'"~' ax'" ' ' ax 

where t,, T3, ..., Tm-s are holomorphic functions of ic in the vicinity 
offl!=0. If then 

Tixfy^xPBix.p), 

the indicial function of 2' is the coefficient of x" in $ (x, p), which 
is a polynomial in p and coutains no negative powers of x. This 
coefficient may be independent of p ; in that case, the character- 
istic index of 2" is m — 5. Or it may be a polynomial in p, say of 
degree k in p, where k^0\ the characteristic index of T then is 



i N' = TS, the indicial function of N is the product of 
the indicial functions of T and S; so that the ifidicial /unction of 
S, which gives all the regtdar integrals of N, is a factor of the 
indicial function of the original equation. The degree of the 
indicial function of 8 is equal to s, because S = is an equation of 
order s of Fuchsian type ; the degree of the indicial function of N 
is m— w, where n is the characteristic index of ^=0. Hence 



so that (assuming for the moment that k may be either zero or 
greater than zero) an upper limit for the number of regular 
integrals which an equation can possess is given by 



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77.} THE NUMBER OF REGULAR I^TEGRALS 231 

where m is the order of the equation, and n is its characteristic 
index (supposed to be greater than zero). It is known that, when 
n=0, the number of regular integrals is equal to m. 

Corollary I. An equation, whose indicial function is a 
constant, so that its indicial equation has no roots, has no regular 
integrals; for its characteristic index is equal to its order. But 
such equations are noC the only equations devoid of regular 
integrals. 

CoROLLABY II. When k is equal to zero, then s is equal to 
m — n, so that the number of regular integrals of the equation 
is actually equal to the degree of the indicial function. The 
necessary and sufficient condition for this result is that the 
equation, which is reducible, must be capable of expression in 
the form 

where the indicial function of ?■ is a constant, and tho degree of 
the indicial function of S is equal to the order of S. 

This result, which is of the nature of a descriptive condition, 
appears to have been first given in this form by Floquet*. Other 
forms, of a similar kind, had been given earlier by Thom^f and by 
FrobeniusJ (see | 83, post). 

Note. On the basis of the preceding analysis, it is easy to 
frame an independent verification that the characteristic index 
is not greater than m — s. For in the operator T, the quantity 
Tm^s^ic does not vanish when a: = ; and all the quantities r>, , such 
that 

\<m~ s — k, 

do vanish when a: = 0. Hence, when we take iV as expressed in 
the form 

the coefficient of 

is the first (in the succession from left to right) in which Tm„(_i 
occurs; it also contains q^, t,, .... Tm-s-s-i, all of them occurring 

* Ana. de I'^o. Nona. Sup., 2° S6r., t, vui (1879), Buppl, pp. 63, 64. 

t Crelle, t. lxxvi (1S73), p. 286, 

t GrelU, t. isns (1875), pp. 331, 832. 



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232 NUMBER OF REGULAB INTEGRALS [77. 

linearly. When a;=0, all of these except t™„j_j: vaoish, and 
Tn-t-n does not vanish ; and therefore qm-a-i does not vanish when 
a!=0. In the coefficient of 

of- -, — , 

where iJ.>s-\-k, the quantities i/o, ti, ..., t^_„ occur linearly: each 
of these vanishes when ic = 0, and therefore jm-» does vanish when 
ic = 0. As this holds for all values of /i, it follows that (jms-k is 
the first of the quantities q which does not vanish when ic=0; 
hence the characteristic index of N is in — b — k, that is, it is 
^m—s, where s is the number of regular integrals possessed by 
the equation iV = 0. 

E<c. 1. If w=Mii be an integral, regular and free from l<^arithm8, of an 
equation P=0, which is of order m and haa i regular integrak, and if a new 
dependent variable u be given by 

ahew that u satisfies an equation Q = 0, which is of order m - 1 and has s — 1 
regular integrals ; and obtaiQ the relation between the characteriatic index of 
P=0 and that of §=0. (Thomd.) 

Ex. 2. The equation 



s integrals regular in the vicinity of - = and linearly independent of 
th d=0 lltjJ P hwthtt pl(t 

1 1 t ) f ea h f th m g uefii t p (Tl m ) 



b ti^ ly as gn d bj t t th d t th t =0 pi 

d yptp thttl m wffi i p be d t n ed 

a. t perm tth qt tp ss — bt lyaasged gila, 

t gr 1 h ealj d p d t f th (rhm^) 

^4P thtth dt ydflitt tht 

luftt \=0 f i d 1 vmg d If t i d f, 

yhllh yilealydpdt-^l t 1 thtV 

lllbepltfthfm QMD wh-eth d If i la i Q M D 

are of degrees 5, 0, m-y-h respectively, and D is of order m-y-S. Is 

there any limitation upon the order of Mf (Cayley.) 

Ex. 5. Shew that an equation QD=0 has at least as many regular 
integrals as Z> = 0, and not more than Q = and 5 = together ; and that, if 
all the integrals of D=0 are regular, then QI>=0 lias as many regular 
integrals as §=0 and i)=0 together. 



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71.] AKD DEGREE OF INDICIAL FUNCTION 233 

Hence (or otherwise) shew that, if an equation F=0 has all its integrals 
regular, then F can be resolved into a product of operators, each of the first 
order and such that, equated to zero, it has a regular integral. la this 
resolution unique? (Frobenius.) 

78. In the two extreme cases, first, where the degree of the 
indicial function is equal to the order of the equation, and second, 
where its degree is zero, the number of regular integrals is equal 
to that degree. The preceding proposition shews that, in the 
intermediate cases, the degree merely gives an upper limit for the 
number of regular integrals. It is natural to enquire whether 
the number can fall below that upper limit. 

As a matter of fact, it is possible* to construct equations, the 
number of whose regular integrals is less than the degree of the 
indicial function. Taking only the simplest ease leading to equa- 
tions of the second order, consider the two equations 

U = ^ + ky + h = 0, F=J + % = 0, 
da! " dx " 

of the first order ; and form the equation 

ax ax 

which manifestly is of the second order, say 
d'y dy 



If we can arrange so that ic = is a pole of p of order n, where 
n ^ 2, then « = in general will be a pole of q of order ;i + 1 ; and 
the indicial function will then be of the first degree. 

Consider now the equation of the second order. Since 
0"- V^-h, 



it can be written 



which is satisfied by 



i — - 

dx 



where A is any ai'bitrary constant. 



V (1872), pp. 311—313. 



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234 THOMfi's THEOHEM [78. 

Let Y be an integral of the equation of the second order. It 
may be an integral of F = ; if it is not, then, when we take 



that is, 1/, is an integral of U= 0. Thus any integral of the 
equation of the second order either is an integral of 1^= or is a 
constant multiple of an integral of ^7=0. If, then, U=0 and 
V=0 are such that they possess no regular integral, the differ- 
ential equation of the second order can possess no regular integral ; 
at the same time, its indicial function is of the first degree. 
The equation V"= will not have a regular integral, if a; = is a 
pole of k of order greater than unity ; and the equation U=0 will 
then not have a regular integral, if /i is a rational function of x. 

Ex. 1. The aggregate of conditions can be aatiisfled aimnltaneously in 
many ways. For instance, take 



'-s+i. «- 



Z+^x 



-s-i; 



The differential equation of the second order is 

d^y 1_ c^_3 + 2^_, 

dx' ^ dx 

its indicial equation is of tlie first degree, and it has no regiilar integrals : or 
the number of its regular integrals is less than the degree of its indicial 
equation. 

The conclusion can otherwise be verified ; for it is easy to obtain two 
linearly independent integrals in the form 



1-7^ 



no linear combination of which gives rise to a regular integraL 
Ex. 2. Shew that the equation 

has no regular integrals : and verify the result by obtaining the integrals of 
the equation, (ThomiS, Floquet.) 



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79.] DETEBMINATIOS OF THE REGULAR INTEGRALS 



Determination of such Regular Integrals- as exist. 

79. When the degree of the indicial function of an equation 
of order m is less than m, no precise information is given as to the 
number of regular integrals possessed by the equation. The further 
conditions, sufficient to determine whether a regular integral 
should or should not be associated with any root of the indicial 
equation, can be obtained in a form, which is mainly descriptive 
for the equation of general order and can be rendered completely 
explicit for any particular given equation. 

Let the equation be 



A {w) = go^'" ^^ + q,x"^' -^^^ + . . . + <;™w = 0, 




of characteristic index n. Let E{0)he the indicial function 


, and 


let a be one of its zeros, so that 





Then, if a regular integral is to be associated with a-, it n)ust be of 
the form 

w = a^(c„+ c,« + Csa;^+ ... + CpX^ + ...). 

This expression, when substituted in the equation, must satisfy it 
identically, so that, after substitution, the coefficient of «"+*■ must 
vanish for every value of p : and therefore 

where the number of terms in this difference -relation depends 
upon the actual forms of g,,, q,, ..., 5™. Of the coefficients /|„/i, 
.■■,fr, the first is 

which is of degree m — n in p; of the remainder, one at least, viz. 
fg-m, is of degree m in p, where g has the same significance as in 
§76. 

The successive use of this difference -relation, together with the 
equations for the earlier coefficients, the first of which is 

leads to the values of all the quantities (;^h-c„, for the successive 
values of fj. ; and thus a formal expression for it is obtained that 



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23C DETERMINATION OF THE [79. 

satisfies the equation. If, however, the expression is an infinite 
series, it haa no functional significance when it diverges : that this 
frequently, even generally, is the case, may be inferred as follows. 
For if c^+i -i- c^, with indefinite increase of (i, tends to a limit that 
is not infinite, so also would C,i-|-a-i-c„+i, C/^^^c,^^, and so on; 
and therefore 

for finite values of o, also would tend to a limit that is not infinite. 
Now a number of the quantities 

7.W' 

for various values of 6, undoubtedly tend to zero as jj, increases 
indefinitely ; some of them may have a finite limit ; but one at 
least is infinite, viz. 

/.w ■ 

because the numerator is of degree n higher than the denominator, 
both of them being polynomials in /i. Consequently, the ex- 
pression 

acquires an infinite value as i* increases without limit. The 
difference -relation requires the value of the expression to be 
always - 1, so that the hypothesis leading to the wrong inference 
must be untenable. Therefore c,^i-i- c^, with indefinite increase 
of /i, does not tend to a limit that is finite, and therefore the 
series diverges*. There is then no regular integral to be asso- 
ciated with the root a. 

* It is not inconceivable that, for special values of in and of n, and for special 
forms of the eoefEoients q, as well as for a speoial value of the limit c„4.j4-Cji, the 
infinite parts of the expression 

,J,/7W "•, 
might (lisappear, and the espression itself be eiiual to -1. In that case, the 
series would oonvei^e : and an exception to the general theorem would occur. But 
it is dear that such an eioeption is of a yery special character: it will be left 
without further attempt to state the conditions explicitly. 



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79,] REGULAR INTEGRAIS THAT EXIST 237 

As the series thus generally diverges when it contains an un- 
limited number of terms, the regular integral is thus generally 
illusory. The only alternative is that the series should contain 
a limited number of terms : and then the regular integral would 
certainly exist. Accordingly, let it be supposed that the series 
contains k + 1 terms, so that 

Ci Ci Ct 



are quantities known from the difference-relation, and that 

Ck+i, c*+2. ■■■ ad inf. 
all vanish. If we secure that cj+i, ct+i, ..., Cn+r all vanish, then 
every succeeding coefficient must vanish in virtue of the diffei'ence- 
relation ; and these t relations will then secure the existence of a 
regular integral to be associated with the exponent cr. Taking 
p — k, k~l, ..., k — r + 1 in succession, we find the t necessary 
conditions to be 

/„ (k) Ci = 0, that is, /„ (k) = 0, 
and generally 

for values r= 1, 2, ..., t- 1. The first of these is 

/S(a- + k) = 0. 
so that the indicial equation, which possesses a root er, must 
possess also a root <T + k, where k is a positive integer. (In the 
special instance, when k = 0, no condition is thus imposed : in the 
general instance, when & is a positive integer greater than zero, it 
is easy to verify that E{a- + k) is the indicial function for ic = x .) 

When the aggregate of conditions, which will not be examined 
in further detail, is satisfied in connection with a root of the 
indicial equation, a regular integral exists, belonging to that root 
as its exponent; and there are as many regular integrals, thus 
determined, as there are sets of conditions satisfied for each root 
of the indicial equation. 

Explicit expressions for the various coeOicients c can be derived, 
when the explicit forms of the quantities q are known : but the 
general results involve merely laborious calculation, and would 
hardly be used in any particular case. The results are therefore. 



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238 MODE OF OBTAINING [79. 

as already remarked, mainly descriptive : and so, in any particular 
case, it remains chiefly a matter for experimental trial {to be 
completed) whether a regular integral ia necessarily i 
with a root of the indicial equation. 

For this purpose, and also for the purpose of c 
regular integrals associated with a multiple root of the indicial 
equation, a convenient plan is to adopt the process given by 
Frobenius (Chap. Ill) when all the integrals are regular. We 
substitute an expression 

W = CoClf -It CiiC^+'+ ... +c^af-^i^+ ... 
in the equation 

of characteristic index n. After the substitution, the ttrst term is 

where E{p') is the indicial function, of degree in — n; and we make 
all the succeeding terms vanish, by choosing the relations among 
the constants c appropriate for the purpose. We thus have 

N{w) = cE{p)a^; 
and the relations among the constants c are of the form 

where the constants ix^^^-i' ■■■> '^co ^^'"^ polynomials in /i and, 
when this relation is the general difference- relation between the 
coeiBcients c, one at least of these polynomials a^r is of degree m 
in II. When the difference -relation is used for successive values 
of fj,, we obtain expressions for the successive coefheients c, which 
give each of them as a multiple of c^ by a quantity that is a 
rational function of fi. When these coefficients are used, we have 
the formal expression of a quantity vj which satisfies the equation 

Unfortunately for the establish men t of the regular integrals, this 
formal expression does not necessarily (nor even generally) con- 
verge: for, in the difference-relation among the constants c, the 
right-hand side is a polynomial of degree m in [i, while the left- 
hand side is a polynomial of degree m — m in /i, so that the series 

Se^a^''''' 
would, as in the preceding investigation, generally diverge. 



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79.] EEOULAR INTEGRALS 239 

But while this is the fact in general, it may happen that the 
series would converge when p acquires a value occurring as a root 
of the equation 

£(p) = 0. 

In that case, the series satisfies the equation 

if(»)-0; 

in other words, it is a regular integral of the differential equation. 

Further, if the particular value of p be a multiple root of the 

indicial equation, it can happen that the series 

div 

Tp 

converges for this particular value of p ; and then 
^g).l|o.«(p).-) 
= 0, 
because the value of p is a multiple root of E= ; in other words, 
p- is then a regular integral of the differential equation. And so 

possibly for higher derivatives with regard to p, according to the 
multiplicity of the root of E=0. 

The whole test in this method is therefore as to whether the 
series 

converges for the particular value (or values) of p given as the 
roots of the indicial equation. The method of dealing with a 
repeated root of the indicial equation has been briefly indicated. 
Corresponding considerations arise, when £" = has a group of 
roots differing among one another by integers. In fact, all the 
processes adopted (in Ch. iii) when all the integrals are regular, 
are applicable token only some of them are regular, provided the 
various series, whether original or derived, are converging series. 
The deficiency, that arises through the occurrence of divei^ing 
series, represents the deficiency in the number of regular integrals 
below m — n. As already stated, the tests necessary and sufficient 
to discriminate between the convergence and divergence of the 
various series are not given in any explicit form, that admits of 
immediate application. 



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240 EXAMPLES [79. 

£x. 1. Consider the equation 

constructed in § 78, Ex, 1. The indieial equation is 

p-3=0, 
BO that there ia not more than one regular integral ; if it esists, it belongs 
to an exponent 3. To determine the existence, we substitute 

in the original equation ; that it may he satisfied, we must have 

0=^{(» + 2)(™ + l)-2}c„_, + nc„, 
for « = 1, 2, .... We at once find 

and therefore 

The series S c„a-^*" diverge^, and tlierefoie the one possible le^ulii lute^iil 

does not exist; that is, the oiigmal equation po^isc'ises no leguUi intPoHl, 
although the indieial equation is of the first degiee 

If there were a regular integral, it would satisfy an equation 

where w is a holomorpbic function of ^ ; and the original equation could then 
be written 



(''£-) ('i-')-. 



where » is some holonsorphic function in the vicinity of x-=0. It might be 
imagined that, as the indieial equation is of d^ree unity (a property that 
does not forbid the existence of a regular integral), it would be possible to 
obtain the regular integral through a determination of u, and that the 
divergence of the series in the preceding analysis is due to the operator 



which annihilates only expressions that are not regular. That this is not the 
case may easily bo seen. We have 

so that, if the resolution be possible, we have 



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79.] EXAMPLES 241 

Substituting in the second of these the value of v given by the first, we find 

as an equation to determine «, supposed a holomorphio fuiiotion of x. Let 
>>e substituted ; in order that the equation for v, may be satisfied, we have 

and, for values of m higher than zero, 

(K + 2ao-l)ii, + a(a^a,_i+asa,._s+...)-a„ + ,=0. 
Hence Oo=3, a,=4, a^=2i, and so oo. The relation giving «„+„ when taken 
for successive values of n, shews that all the coefficients a are positive ; hence 

>(« + 5)<i„, 
that is, 

30a„>(™ + 4)!, 

and so the series for m diverges : in otber words, there is no function m, and 
the hypothetical resolution of the equation is not possible. 

Mole. This ai^umont is general ; it does not depend upon the particular 
coefficients for the special equation that has been discussed. 

Sx. 2. Consider the equation 

which is in the normal form. The characteristic index is 1 ; the indicial 
equation is 

(9(fl-l)-5d+9=0, 

{fl- 3)2=0, 
BO that the number of regular integrals cannot be greater than two, and such 
as exist belong to the exponent 3. 

To detei'mine these regular integrals (if any), we adopt the Frobenius 
method of Ch. Iii, Taking 

^=c„xP + c,afi*'^ + ... + c^sf + " + ..., 

provided 

_ p2-2p-5 
■^1 '''* p-2 ' 
and, for values of n greater than unity, 

= (p + «.-3)c,+{p^ + p(2n-4) + )i2-4H-2}c„.i--2(p + n)c„_a, 
a factor p + m — 3 having been removed, because it does not vanish for these 
values of ?i. Let 

(p+»-3)e,-2«„_, = 4„. 



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242 DETERMINATION OF REGULAR INTEGRAL [79. 

X;. = (p-2)<!,-3o„=-(p-3)(p + l)c„. 
Also the difference-equation for the coetKeients c becomes 

i:,=l^l)'->{p + n)ip + n~l)...{p + 2)k, 
=.(_!)» (p_3){p + l)(^ + 2)...{p + ™)c„ 

Hence, writing 

2" 
"""nO. + ft-S)'*'' 
in the rtslation 

and substituting the value of i,„ we have 

Adding the aides of tllis equation, talsen snccessiveiy for m, b-1 3, 2, and 

noting tliat 

.J{5 + »p-i>")n()>-3)t„ 
...,.[M5+2,-,')n(p-3) + (,-3)J_(-i)."Jsi»yiisi"^]. 

We thus have a value of y in the form 

where 

S (-irn(p + m)n(p+m-4) 
:-^2"-.(5 + .p-p^) 'V;^-- +(p-3)^" 



n(p-3) _ _ 

nMn(p+»-a) 



and this satisfies the relation 

DV=c„{p~3fxi: 
It is clear that formal solutions of the original differential equation ai 



'^- [f],. 



Of these, the first is 

in effect, a constant multiple of sfl^ ; and the second is 



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79.] WHEN IT EXISTS 243 

because a series, in which 

is the coefficient of j^*^, manifestly diverges*. 

It thus appears that, although the indicial equation for x=0 is of the 
second degree, the differeutial equation possesaes only one integral which is 
regular in that vicinity ; and this integral is a constant multiple of x^e^. 

This regular integral satisfies the equation 

«*-(3+»),.0, 

SO that the original equation must be reducible. It is easy to verify that it 
can be expressed in the form 



, + .|-(3 + 2.)}{.*-(3 + 8.);,}.0. 



JH.v, 3. As an example which allows the convergence of the series for the 
regular integral to occiur in a diffei'ent way, consider the equation 
^y'_(l_2a;4-2a^)y+(I-2jr;+;i;2)y=0. 

The indicia! equation is 

P=0, 

so that one regular integral may exist. To determine whether this is so or 
not, we substitute 

which (if it exists) belongs to the exponent aero. Comparing coefficients, 



and, for all values of n that aro greatei' than unity, 

(B + I)«„^.i = (»^ + m + l)o„-3aa„_i + a„_j. 
Let 

In general, the values of a (and the consequent values of a) as determined by 
the last equation, lead to diverging series ; but in our particular case, 



o that 0^=0, 0^ = 0, and generally Cm^O, that is, 



• The series in ijg is saved from divergence because, in it, these ooeffioieiits are 
muldplioii by the taotor p - 3, which vanishes for the special value of p and whidi 
therefore removes the quantities that cause the divergence in the Eecond. integral. 



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244 

and therefofe 



EXISTENCE OF 



[79. 



provided 



so that a regular integral esiats. It is a constant multiple of e'. 
Ex. 4. Consider the equation 
i) (J.) = (aH5 4- ^k") y"" + (^ + 4iK«+ i!*) y'" - (2ii72 + 3j!»+ ar«) y 

+ (3ar + toH 4iJ^) y - (3 + to + 4a;S) ,?/ = 0, 
The characteristic index is unity ; hence the numher of regular integrals is 
not greater than three. To determine them, if they exist, we take an 
cxpi'eBsion 

and form 2)(y), choosing relations among the coefficients c such that all 
terras after the first in the quantity D (y) vanish. We thus find 

Cip2(p-2) + c„(p-l)(p-2)(f,Hp-3) = 0, 
and, for values of n greater than unity, 

t.(p + »-ir'(j. + »-3) + <.-,(p + — 2)(p+«-3)& + »)'-(p+«)-3) 

+ «.-,(|, + l>-S)'(p + »-4)(p + »)_0. 
The indkial equation is 

(p-l)'(p-3).0 

of degree 3 aa was to be expected ( = 4 — 1), be use tl 
is 1, The roots form a single group; ifaregulir t o 
the root 3, it wiii be free from logarithms ; f tw eg 1 
belonging to the root 1, one of them may or may t b fr t 
and the other will certainly involve logarithms. 

Consider the root p=3. As p+K-3 thei nish f 
values of n, wo may remove it from the differen qu fc 

<^„(p+«-l)'' + o.^,(p + »-2){(p + ™)=-(p + «)-3} 

c,(p + m-l)Hc,_,{p + m-2)(p + »-3) = *„, 
we at once find 

i„ + (p + )i)i„_i = 0. 

We require the value of Jc^. We have, for p = 3. 



til gt 

i- m 1 g thm 



J th t th 1 tt 



Taldng 






*a = 1602 + 6e, = 80Co. 
i„ = (-l)''(« + 3)(»i + 2)...6ij 
= §(~l)"(«.-i-3)!<r,; 



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79.] REGULAR INTEGRALS 

ao that, writing 

Aa «| and «a are positive, it follows that all the 
and clearly 

80 that the series 

divei^s ; and there iis no regular integral belonging to the root 3. Moreover, 
the coefficient of o„, being (p+n—l)\ does not vanish when p=3 for any 
value of TO ; hence, if two regular integrals exist belonging to the root unity 
of the indicial equation, one of them will certainly be free from logarithms. 

Consider now the repeated toot p = l. As p + )i — 3 vaniahes for this 
value of jj when «.=2, the difference-equation is then evanescent for ji = 2 and 
it does not determine r^. For other values of n, the quantity p4-«-3 does 
not then vanish, so that it may be removed. We then have, for values of 
m ^ 3, the same form of equation as before, viz, 
■^„(p + m-I)^ + c„-i(p4-ft-3){(p + »t)^-(p + ™)-3} 

+ c„_,(p-f»-3)(p + «-4)(p+«) = 0. 
Also 

c,= -(p-I)(pS + p-3)^, 

the value p = l not yet being inserted because we have to differentiate with 
regard to p. Tlie difference-equation for jj=3 gives 

9(3-1-1802 = 0, 

For values of n.^4, let p = iT-2, so that the value of o- is 3 ; take m-2=m, 
ao that the values of m are ^ 2 ; and write 

then the difference-equation becomes 

6,„(^+m-l)'-f&™-.(<r + m-2){(<r-t-w)^-(ff + i«)-3! 

-H6«_2(o-t-m-3)(o-Hm-4)(o-l-m) = 0. 
Here <r=3, m^2; e2=&o, C5 = 6[--: 
exactly the same as in the former case 

,^[b^-a^x+a % - ) 
with tl o cailier notation it certainly diverges inles^ 6j|=0. If 6=0, every 
ciefecient vanishes, and the senes itself vanishes As we require regular 
mte^rils we shall therefore assume 6^ = that is, i.;=0; and then all the 
remaining c effiuenta \inisli so that we h \e 

1= .[■^- ■'"^'(p-l)(p'+p-3)^], 



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246 EXAMPLES [79. 

au osprossion which is such that 

are integrals of the equation 
The former ia o^s : one regidar integral thus is 
The latter ia 

another regular integral in 

The original differential equation accordingly has two regular integrals. 
S!x. 6. Shew that the equation 

haa one integral regular in the vicinity of :i;=0-, and express the equation in 
a reducible form. 

Sx. 6. Shew that the equation 

has two regular integrals in the vicinity of x=0, in the form 

e", x^ ; 
and obtain the integral that is not regtilar. 

Ex. 7. Shew that the equation 

has no integral, that ia regular in the vicinity of m=<i ; ej[preas the equation 
in a reducible form, and thence obtain the integral by quadratures. (Cayley.) 

Ex. 8. An equation P — can be e.^pressed in the form 
$0 = 0, 
where ^=0 has no regular integrals ; can P=0 have any regular integrals ? 
Illustrate by a special case. 

Ex. 9. Ill the equation 

the coefficients P are poljDomials in a: of degree p, and p<.n: shew that it 
possesses n—f integrals, which are integral fuuctions of a;. (Poincar^.) 



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lUKElJUCIELE EQUATIONS 



Existence oe laREUUCiBLE Equations. 

80, We have seen that an equation is reducible when it ia 
satisiied by one or more of the integrals of an equation of lower 
order, in particular, by the integral of an equation of the first 
order. The main use so far made of this property has been in 
association with the regular integrals of the equation: but it 
applies equally if the equation possesses non-regular integrals 
that satisfy an equation of lower order. It is superfluous to 
indicate examples. 

It must not be assumed, however, that every equation is 
reducible by another, if only that other be chosen sufiiciently 
general. On the contrary, it is possible to construct an irre- 
ducible equation of any order m, as follows*. 

We construct an appropriate characteristic function which, as 
is known (§ 76), uniquely determines the equation. Take a poly- 
nomial in p of degree m, say 

let the coefficients of the powers of p be holomorphic functions of 
«, not all vanishing when « = 0; and let the function, subject to 
these limitations, be so chosen that, when arranged in powers of 
X in the form 

h {^., p) = h, (p) + xh, (p) + a;'h,(p} +..., 

ho (p) is independent of p and not zero, and h^ (p) is of degree m in 
p. Then if iV=0 is the equation determined by h(x, p) as its 
characteristic function, JV= is irreducible. 

Were N reducible, an equation S = of lower order s would 
exist such that each of its integrals satisfies N = <i; and then an 
operator Q, of order m — s, could be found such that 

N=QD. 

Wc take Q and D in their normal form ; and so N is in its normal 
form. Now 

Q{af) = af {^,(p) -(- a^{p) +,af^,ip) + ...[, 



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248 REDUCIBILITY OF EQUATIONS [80. 

the right-hand sides of which are polynomials in p of degrees m — s 
and s respectively. Then, as in § 76, we have 

K ip) = ?„ ip) vi (p) + ?. (p) v« ip + 1)- 

Now /(„ (p) is a constant, being independent of p ; hence, owing to 
the polynomial character of Q {w") and D (a^) in terms of p, the 
two quantities ^((p) and ijoCp) £"^e constants. Accordingly, ijo{p + 1) 
is a constant ; and therefore the degree of 

f.</>)i.(p)+f.(p)i.((>+i) 

in p is the degree of 171 (p) or fi (p), whichever is the greater. But 
the degree of i)i(p) is not greater than s, and that of ^i{p) is not 
greater than m — s; so that, as s > 0, the degree is certainly less 
than m. But the espresaion is equal to A, (p), which is of degree 
tn. Hence the hypothesis adopted is untenable ; and the equation 
N=0, as constructed, is irreducible. 



Equations having Regular Integrals are Rgducible. 

81, Suppose now that, by the preceding processes or by some 
equivalent process, the regular integrals of the equation N=0 
have been obtained, s in number, and that the equation of which 
they constitute a fundamental system is S = 0, of order s: a 
question arises as to the other m — s integrals of a fundamental 
system of N= 0. Let 

where T and S (and therefore also iV) are taken in their normal 
forms. The s regular integrals of if, say y,, y^, ..., ijs, all satisfy 
iS = ; and no one of the m—s non-regular integrals of N, say 
W], Wj, ..,, Wm—s. satisfies jS = 0, for this equation has all its integrals 
regular. Let 

S(»,) = »„ (.-=1 m-,); 

then, as N{wr) = 0, we have 

r(«,) = o. 

Now Wr is not a regular expression; hence n^ is not regular, 
that is, it contains an unlimited number of positive and negative 



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81.] HAVING HEGULAK INTEGKALS 249 

exponents when it is expressed as a power-series. Accordingly, 
the m — s quantities u are integrals of the equation 

which is of order m — s and has no regular integrals ; and the m~s 
non-regTilar integrals of iV= are given by 

S (»,) = «,. 
it being sufficient for this purpose to take the particular integral 
and not the complete primitive of the latter equation. 

The case which is next in simplicity to those already discussed 
arises when s = ni — 1, so that the original equation then possesses 
only one integral which is not regular. The equation T^O is 
then of the first order. 

With the limitations laid down, the normal form of T is 
d 

where q, and q, do not become infinite when a; = 0. As the integral 
of T(u) = is not regular, it follows that ^i does not vanish and 
that g,, does vanish when a:= 0; so that, if 

where a is a positive integer > 1 and Q {x) is a holomorphic 
function in the vicinity of « = 0, sach that Q (0) is not zero, the 
equation determining u is 

1 ^ a. A_ _9i - 

say 

lg + ^i + ?i<^ + ... + ? + iiW-o, 

where iJ (a:) is a holomorphic function of x in the vicinity of iC = 0. 
This gives 

%+ -»i, + -.. + --' 
u-ar-e' ' -P.W, 

where Pi is a holomorphic function of x in the vicinity of a: = ; 
and then to determine w, the non-regular integral of N—0, we 
need only take the particular integral of 



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250 REDUCJBTLITY [81. 

where 

in which qa, qi, ..., Jm-i denote lioloraorphic functions of x, and ^o 
does not vanish. Writing 



the equation for v takes the form 



where qo, p,, p^, ..., pwt-i are holomorphic functions of «, such that 
5o and pm-i do not vanish when iC = 0. 

In some cases it happens that a particular integral of this 
equation exists, in the form of a converging power-series repre- 
sented by 

«'— "— -PW. 

where P{x) is a holomorphic function of a;: in each such case, the 
non-regular integral of the original equation is 

«■"'— «°-PW. 

But, in general, the particular integral of the u-equation is not of 
the same type as the regular integrals of the original equation : 
and then the non-regular integral of the preceding equation 
cannot be declared to be of that type. 

Ex. An illuatration is furniahed by the equation in Bs, 6, § 79, viK. 

It has two regular integraJa, viz. 

and these constitute the fundamental system of 



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81.] THE ADJOINT EQUATION 23 

in the normal fonn. To have the given equation in the normal form, v 
multiply throughout hy x'' \ and then it must be the same as 

when f is properly determined. We easily find that 
and so the equation for determining m, where 
y being the non-regular integral, is 

v.<ix~ 3?{\■\■'i^•\■%a?^^a^) 



ao that 

Hence the non-regular integral of the original equation <ai'i 
particular integral of 

„ ^ , 1-(-2j; + 3j;3 + :k* \ 
f-y^y^— -^ 1". 

Let j/ = v^ ; the equation for v ia easily found to be 



"(-i) 



satisfied by v = l : and therefore the non-regular 



The Adjoint Equation, and its Properties. 

82. Of the properties characteristic of a linear equation, not 
a few are expressed by reference to the properties of an associated 
equation, frequently called Lagrange's adjoint equation. It is a 
consequence of the formal theory of our subject, as distinct from 
the functional theory to which the present exposition is mainly 
limited, that Lagrange's is only one of a number of covariantive 
equations associated with the original. As its properties have been 
studied, while those of the others remain largely undeveloped. 



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252 PROPERTIES OF lagrange's [82. 

there may be an advantage in giving some indication of a few 
of its relations to the original linear equation. 

The latter is taken in the customary form 

P(?^) = P,wi»' +P,wi"-'» + P,w"'-=' + ... +P„«; = 0, 

where w'*"' is the rth. derivative of w with respect to z; and from 
among the various definitions of the adjoint equation, we choose 
that which defines it to be the relation satisfied by a quantity v in 
order that vP(w) may he a perfect differential. Now, on inte- 
grating by parts, we find 

+ ^, (vPr) W l"-^^> -...+(- 1 )"-^ [w ^- {vPr) dz, 

for all the values of r\ hence, writing 

P. =/-•., 



R (w, v) = p„w '""" 4- piW "'~^' + . . . + pa-iW, 

PW = P,.-^(P,_„) + ;|(P,.-..)^.., + (-1)"£(.P.), 

we have 

jvP (w) dz = R (w, v) + iwp {v) dz, 

and therefore 

.P(«.)-«p(.).^(K(»,.)}. 

It is clear that, in order to make vP{w) a perfect differential, 
whatever be the value of w, it is necessary and sufficient that v 
should satisfy 



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82.] 



ADJOINT EQUATION 



a linear equation of order n, commonly called Lagrange's adjoint 
equation ; and further that, if v is regarded as known, then a first 
integral of the equation P (w) ;= is given hy 

B {«,.«) = .. 
a being an arbitrary constant, and R being a function manifestly 
linear in w and its derivatives. 

Further, since 

jwp (v) d^^-R (w, V) + jvP (w) dz, 
it is clear that wp {v) is a perfect differential if 

P(») = o, 

shewing that the original equation is the adjoint of the Lagrangian 
derived equation: or the two equations are reciprocally adjoint to 
one another. 

Ex. Shew that, if ic,, ,.., w„ be a fundamental system of integrals of the 
et[uation i*(w) = 0, then a fundamental system of int^rals of the, adjoint 
equation p{v) = lS is given by 

1 -Ir." 



Shew also that the product of the respective determinants of the two sets of 
fundamental integrals depends only upon 1'^. 

One immediate corollary can be inferred from the general 
result, in the case when the equation P{w) = Q is reducible. 
Suppose that 

p(«,)=p.p,(»)=p,(fl'), 

say, where W^P^iw); then we have 
(vP(w)ds= Lp,(W)dz 

^R,(W,v)+jwP,(_v)dz, 

■where P, is the adjoint of P„ and R^ is of order in W and in v 
one unit less than P,. Again, writing 

F=P,W, 



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254 THE ADJOINT EQUATION [82, 

we have 

where P^ is the adjoint of P^, and iE^ ia of order in V and in v one 
unit less than P,. Combining these results, we have 

jvP(w)dz^R,(W,v)-\-R,{V,v) + jwP,(V)dz 
= R{w,v) + jwP,P,(v)dz, 

where R is of order one unit less than P in w and in v. It follows 
that 

P,P,{v)^0 
is the adjoint of 

P(»)- p.p. (") = !), 
where P, , Pi are adjoiut to one another, and likewise Pj, Pj. 

By repeated application of this result, we see that the adjoint 
of 

P<«/j=P,P,...P,(M.) = 

is given by 

P;P^,...P,P,(i')=0, 
Hence the adjoint of a composite equation is compounded of the 
adjoints of the factors taken in the reverse order. Manifestly 
an equation and its adjoint are reducible together, or irreducible 
together. 

The expression R (w, v) is linear in the derivatives of w, up to 
order n — 1 inclusive, and also in those of v, up to the same order : 
it may be called the bilinear concomitant* of the two mutually 
adjoint equations. 

For further formal developments in respect to adjoint equa- 
tions and the significance of the bilinear concomitant, reference 
may be made to Frobeniusf, Halphonj, Diui§, Cels|], and 
Darbouxt. 

* Begleitender hilinearer Differentialaitsdmclc. with Frobeniua, 

t CreUe, t. lxixv (167S), pp. 1S5— 213; referencea ace given to other writers. 

t Liouville'a Journal, i' S^c, t. i (1885), pp. 11—85, 

§ Aim. di Mat., 3» Ser., t. n (1899), pp. 297—324, ib., t. ni (1899), pp. I2S— 183. 

II Ann. de VEc. Norm., 3' Sir., t. viii (1891), pp, 341^15. 

IT Theorie generale dee surfaces, t. ii, pp. 99 — 121. 



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82.] EXAMPLES 255 

Ex. 1. Prove that, if ii linear equation of the second order ia self-adjoint, 
it is cspressible in the form 

that if a linear equation of the third order, in the form 
ia effectively the same as its adjoint equation, then 



and find the conditions that a 
self-adjoint. 



r equation of the fourth order should be 



Ex. 3, Prove that, if the equations 

yoHW-i-nyiVi""'!-!- "Yi — y.j»ff"~^4----=0, 



e adjoi 



it to one another, then 
ri= -ffi+ffo'y ^1= -7i + To'' 

73= -ffs + ^ffi-^ffi"+go"\ ?3= -■ V3 + 3y2■-■V + 7o"'> 



and obtain the expression of the bilinear concomitant. 



i„ denote any « arbitrary functions of x, such that 















dx^-'-' 



does not vanish identically; and suppose that these functions of x are 
regular in a given region of the variable, aa well as the coefficients a of the 
etjuation 

Further, let a set of quantities p bo constructed according to the law 

dpn dp, dv„_, 

»-"•• ?■-"<-£. »-'".-j;f' ■■•■ '••-'•---^nr- 



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256 PROPERTIES OF laorange's [82, 

and let the last of them be denoted by - Z, so that there are n functions Z 
con'osponding to the n functiona z. Shew that, if Q (c) is the value of § when 
the last column of the latter iiS replaced by constants c„ ..., c„, if §{a', a\) is 
its value when the last column is similarly replaced by Sj (a:,), %(a:i), ..., 
j„ (x^, and if Q {x, a^i) is its value when the last column is similarly replaced 
by 2i (:Kj), iTj (^cj), ...,Z„(a;,), then 

where a is a value of 3! within the given ri^ion and the constants c are 
determined in association with a. 

Indicate the form of this result when z^, ..., 2„ are a fundamental system 
of the equation, which ia the adjoint of the left-hand side of the above 
equation. 

Also shew how, in even the most genera! case, it can be used as a formula 
! to obtain an infinite converging series of integrals as an 
1 for y. (Dini.) 

83. Consider an expression Piw) and its Lagrangian adjoint 
p (v), and let R (w, v) denote their bilinear concomitant ; then 

,P(»)-»;,(rt.^j(2i(»,„)), 

which holds for all values of v and w. Accordingly, let 

w = 2~''~''~^, V — z", 
where s ia any integer ; then 

^p (,-,—.) _ ,-,-.-,^ (^) _ ^ ^M(z->—'. ^-)). 

Now the left-hand side is a series of powers of z, having integers 
for indices ; as it is equal to the right-hand side, which is the 
first derivative of a similar series of powers, the left-hand side 
must be devoid of a term in s"'. 

Let 

Piz% =SMt)z^^-, 

be the characteristic function of P {w) ; then the coefficient of z~^ 
in scP {2-^-'-') is /s (- p - 5 - 1). Further, let 

be the characteristic function oip{v); then the coefficient of 3~' 
in s"*"-^'^ (s**) is 4>»{p)- Hence 

>^,{(>)=M-p-s-n 



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83.] ADJOINT EQUATION 257 

and therefore 
so that, if 

be the characteristic function of a given equation, then 

S/, (-(>-/" -1)2"' 
is the characteristic function of the adjoint equation. 

When P{w) is in its normal form, all the coefficients /^(p) 
vanish for negative values of fi, but /i,(p) is not zero. Hence 
f^{— p — fi — l) vanishes for negative values of p,, but not 
/ti(~ p ~^)'} and therefore the adjoint expression p(v) is in its 
normal form. Moreover, their indicial functions y, (p), <p^ (p) are 
such that 

/o(/>)=0o(-p-i), Mp)=A(-p-n 

80 that they are of the same degree*, or the characteristic 
indices are the same. Hence if an equation has all its integrals 
regular in the vicinity of a singularity, the adjoint equation also 
has all its integrals regular in the vicinity of that singularity ; for 
the characteristic index is then zero for the original equation, and 
it therefore is zero for the adjoint equation. Similarly, if am 
equation has all its integrals non-regular in the vicinity of a 
singularity, tlie adjcnnt equation also has all its integrals non- 
regular in the vicinity of that singularity ; for the characteristic 
iudes: is then equal to the order of the original equation, and it 
therefore is equal to the (same) order of the adjoint equation. 

On the basis of these two results, we can obtain a descriptive 
condition necessary and sufficient to secure that, if a differential 
equation of order m has an indicial function of degree m~n, the 
number of its regular integrals is actually equal to m — n. 

Let P=0 be the differential equation, with an indicial func- 
tion of degree m — n. Let R=0 be the differential equation 
of order m — n, which has the aggregate of regular integrals of 
P = for its fundamental system ; its indicial function is of degree 
ra-n. Then (§ 75, iv) the equation P = can be < 
the form 

* Thom^, Crelle, t. lssv (1873), p. 276; Frobeiiiua, Crelk, t 



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258 Lagrange's [83. 

where Q is a differential operator of order n. Because the degrees 
of the indicial functions of P and li are equal to one another, it 
follows (from § 76) that the degree of the indicial function of Q is 
zero, that is, the indicial function of Q is a constant, and therefore 
(§ 77, Cor. i) the equation Q = has no regiilar integral. 

Now construct the equations which are adjoint to P = 0, Q = 0, 
R — respectively ; and denote them hy p = 0, ? = 0, r = 0, 
Because R and r are adjoint, and hecause all the integrals of 
R = are regular, it follows that all the integrals of r = are 
regular; and conversely. Similarly, because Q and q are adjoint, 
and because Q = has no regular integral, it follows that q = 
has no regular integral ; and conversely. Further, by § 82, we 
have 

p^rq, 

so that the equation adjoint to P = is 
p = rq = 0, 

and this equation possesses all the integrals of 5 = 0, an equation 
whose indicial function is a constant. Hence it is necessary that 
the equation adjoint to P = should possess all the integrals of an 
equation of order n, having a constant for its indicial function, if 
P = is to have m — n linearly independent regular integrals. 

But this descriptive condition is also sufficient to secure this 
result. For, as the condition is satisiied, we have 

p = rq, 
where the indicial function of q is 2. constant ; hence, with the 
preceding notation, we also have 

P^QR, 
and the indicial function of Q is a constant. Accordingly, as the 
indicia! function of P is of degree m — m, it follows (§ 76) that the 
indicial function of R is of degree m — n; and therefore (Ch. Ill), 
as the order of R — is m — n, all its integrals are regular. But 
P = possesses all the integrals of P = ; and therefore it has 
m~n regular integrals. 

We therefore infer the theorem : — 

In order that an equation of order m, having an indicial function 
of degree m — n, may possess m — n regular integrals, it is necessary 



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83.] ADJOINT EQUATION 259 

and suffia'piit that the adjoint equation should possess all the inte- 
grals of an equation of order n, having an indicia! .function which 
is a consUvnt 

This result was first established by Fiobenius*; and it may 
be compared with the corresponding result obtained by Floquet 
(§ 77). The special case, when w = 1, had been previously discussed 
by Thom^f, who obtained the result that an eqtuition of order m, 
having am, vndiciat function of degree m — 1, possesses m — 1 regular 
integrals, if the adjoint equation has an integral of the form 

/©_!».«-, 

where G(-] is a polynomial in - , and a is a constant. 

We shall not pursue this part of the formal theory of linear 
differential equations further : we refer students to the authorities 
already (§ 82) quoted, as well as to Thom^j, Floquetg, and 
Griinfeldil. 

* Grdle, t. lxks (1875). pp. 331, 332. 

t Crelle, t. lisv (1873), pp. 278, S79. 

X A summary of many of the memoirs a 
Thomfi, published in CreJie's Journal, will t 
pp. 185—281. 

% Ann. de VEc. Norm. Sup., 2' Ser.. t. viii (1879), Supplement, p. 132. 

II CTelle, t. oxY (1895), pp. 328—842, ib.. t. oxvii (1897), pp. 278-290, 
ib., i. cxiii (19tX)), pp. 43-52, 88. 



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CHAPTER VII. 

Normal Integrals; Subnormal Integkai^. 

84. It is now necessary to consider those integrals of the 
differential equation in the vicinity of a singularity, which are not 
of the regular type. Suppose that such an integral, or a set of 
such integrals, is associated with a root 9 of the fundamental 
equation (§ 13) of the singularity which, as in the last chapter, 
will be transformed to the origin by the substitution 

1 

z — a = x, z — -, 

according as it is in the finite part of the plane, or at infinity. 
Let p denote any one of the values of 



then it is known that an integral exists in the form 

where i^ is a uniform function of x in the vicinity of the origin. 
As this integral is not of the regular type, the function ^ will 
contain an unlimited number of negative powers, so that the origin 
is an essential singularity of ^ : in the case of the integrals con- 
sidered earlier, the origin was either a pole or an ordinary point. 
Accordingly, when ^ is expressed as a power-series, it will contain 
an unlimited number of negative powers: it may contain an 
unlimited number of positive powers also, and in that case it has 
the form of a Laurent series. 

Classification of such integrals might be effected in accordance 
with a classification of essential singularities ; but the discrimina- 



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84.] NORMAL INTEGRALS 261 

bion that thus far bas been effected among essential singularities 
is of a descriptive type *, and has not led to functions whose general 
expressions are characteristic of various classes of singularities. 
Accordingly, it ^ possible to choose one function after another 
with differing forms of essential singularity, and to construct 
(where practicable) the corresponding linear equations possessing 
integrals with the respective types of singularity : but there is no 
guarantee that such a process will lead to a complete enumeration. 
There is one such function, however, which is simpler than any 
other, and yet is general of its class. It suffices for the complete 
integration of the linear equation of the first order when the origin 
is a pole of the coetScient ; and an indication has been given (§ 81) 
that it may serve for the expression of an integral of an equation 
higher than the first. The equation of the first order may be 
taken to be 

where a^+'P is a holomorphic function, s being some positive 
integer. Let 

where /' (ic) is a holomorphic function ; then we easily have 
where "^ {x) is a holomorphic function of w, and 

n = 2+i-i+ •■• + -■ 

a!* a^ ' X 

It is clear that ic=0 is an essential singularity of the integral; 
and also that we thus have the complete primitive of the equation 
of the first order. 

It appeared, in § 81 and the example there discussed, that 
such an expression, if not in general, still in particular cases, can 
be an integral of an equation of higher order. 

As all expressions of the form 



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2(i-2 THOMfi'S NORMAL [84. 

where il is a polynomial in - , possess the same generic type of 
essential singularity, we proceed to the consideration of equations 
that may possess integrals of this form. Such an integral is 
called* a normal elementary integral or (where no confusion will 
occur) simply normal. The quantity e", through the occurrence 
of which the point a) = is an essential singularity, is called the 
determining factor of the integral; the other part of the integral, 
being ie<'iy {x) where i^ is holomorphic, is of the type of a regular 
integral, and so the quantity p is called the eivponent of the 
integral. 



Construction of Normal Integrals. 

85. We proceed, in the fii'st place, to indicate Thome's 
method f of obtaining snch normal integrals as the equation 

d™ti! d"^^-!/) dw _ 

die™ '^ dx™~^ ' ' ' "™~^ ^(c "™ 

may possess. (The method gives no criteria as to the actual 
existence of normal integrals: and therefore, if any criteria are 
to be obtained for equations of order higher than the lirst, they 
must be investigated otherwise.) If a normal integral exists, it 
is of the form 



where ii is a polynomial in - ; and il is determined so that, if 

possible, the equation satisfied by u may possess at least one 
regular integral. Lot 

ftc" 
so that 

(,= 1, *, = n', tp^,^tp' + Q.'tp, {p=l,2, ...); 
then 

d"w _ /, , du , d"~'u ^ d"M\ 

dx" \ die dx" ' daf-J 

* Thom^, Crelle, t. sov (1883), p. 75. Cajley, ib., t. c (18S7), p. 286, suggeated 
the came suin-egular ; but the name normal is that nhioh has geiierall; been 
adopted. 

t Crelte, t. lsxvi (1873), p. 292. 



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85.] INTEGRALS 263 

When these quantities, for the successive values of n, are substi- 
tuted in the differential equation for w, the determining factor e" 
can be removed ; and the differential equation for w then is 



d-t. d—'u 


du 


, , (".-D! 


„, , ("'-2)> , 


■)! ' (r-l)!(m-i 


■)!"'"' (r-2)!(>n-i-)!' 




...+(»t-r + l)p. 



forr = l, 2, ...,m. 

If the origioal equation possesses a normal integral, then, after 
the proper determination of fl, the differential equation for u will 
possess at least one regular integral : its characteristic index 
cannot then be greater than m — l, which (after the results in the 
preceding chapter) is a necessary but not a sufficient condition. 

As 12 is a polynomial in «-', its form and degree being un- 
known, let its degree be s — 1, so that s^2; we then have for £1' 
an expression of the form 



il' 



. «£ , «3 , 



Hence in ti, the governing term (that is, the term with highest 
negative exponent of a:) is -^ ; in ij, it is -J^ ; and so on, so that, in 

*„, it is -^. As in § 73, let ot, denote the multiplicity of « = as 

a pole of p, ; then in q^, the governing exponents of its respective 
parts are 

rs, ^, + (r-l)s, ■=r, + (r-2)s, ..., ^r-i + s, ^,. 

Thus the governing exponents in q,. are, so far as they go, less than 
those in q^+i by s, and s ^ 2. Hence, in forming the characteristic 
index for the equation in u, for the purpose of determining whether 
it may possess a regular integral, the governing exponent in q^ 
is certainly greater by s than that in any otiier coefficient; the 
characteristic index is m, the indicial function is a constant, and 
the equation has no regular integral. But, thus far, ii is quite 
arbitrary ; and it may be possible, by proper choice of its constant 



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264 DETERMINATION OF [85. 

coefficients, to secure that a number of the terms in q^ with the 
greatest exponents of *■"' shall disappear. If by thus utilising the 
governing exponent and the constants in SI', we can secure that 
the characteristic inde\ of the equation in u is less than m, the 
indicial function cea-ses to be i con^tint and the equation may 
have a regular intejfral 

In Older that the indicial function may not be a constant, the 
governing exponent of q , must be less than that of q^, by unity 
at the utmost, or that of q,^ must be less than that of q^ by two 
at the utmost, or (tor sjme value of }) the governing exponent of 
qm-r must be less than that of 5 by ? it the utmost ; whereas at 
the present moment, these diminutions are s, 2s, rs respectively, 
where s ^ 2. Hence an initial necessity is that the s — 1 terms in 
qm with the highest exponents of a:'^ shall vanish. Now 

qm,^tm + Pltm-l + ■■■ +Pm~-A+l^m- 

The s — 1 terms in t^ with the highest exponents ol' le"' are the 
same as in II'", because of the form of il' and because 



(but not more than those s — 1 terras are the same) ; hence the 
s~l terms with the highest exponents of (e~^, say the first s — 1 
terms, in 

n'"' +piii''"~' + . . . +p„-jfi' +p„ 

must vanish. 

86. To render this result attainable, it is necessary that the 
greatest exponent must not occur in only a single term of the 
preceding expression, for then the term could vanish only by 
having d, = ; the greatest exponent must occur in at least two 
terms. Consequently no one of the numbers 

ms, ■=T, + (m — l)s, ■5r2 + (m~2)8, ..., ot„, 

may be greater than all the rest, that is, no one of the numbers 

0, ^i~s, W3 — 2s, .... vT^ — ms, 

may be greater than all the rest. Of the quantities 



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86.] NORMAL INTEGRALS 265 

let g be the greatest. Evidently g is greater than unity ; for the 
original differential equation has not all its integrals regular, and 
so CT„ > w for at least one value of n. Now s cannot be greater 
than g\ for any such value would make all the integers in the 
series 

0, ^i — s, «;,— 2s, ..., Wm-ms, 

negative except the first, that is, the first would then be greater 
than all the rest. Hence s^g: and g ^ 2, from the nature of the 



I. When g <2,no value of s is possible; and then there is no 
normal integral of the type indicated. Such a case arises for the 
equation 

when p and q are holomorphic in the domain of ic = and neither 
vanishes when a; = 0. The quantity g is the greater of 1, |, that 
is, it is less than 2 ; so that there is no normal integral. Moreover, 
as the indicial function of the particular equation is a constant, it 
has no regular integral. 

II. When g is an integer (necessarily greater than unity), wc 
manifestly might take s = ^. For two at least of the numbers 



would then be equal to the greatest among them, which is zero ; 
and then two at least of the numbers 

ms, JiTj + (m ~ 1) S, Wa + (jrt — 2)s, ..., ■nr^_i + S, CT^, 

would be equal to the greatest among them, one of these being ms. 

More generally, let n be the characteristic index of the original 
equation, so that 

for all values of /t that are greater than w; then, adding {m—n)(«—l) 
to each side of the inequality, we have 

■OT„ + (m - n) s^w^ + im-ii^s + ifi- n)(s - 1), 

where fi>ii. In the case of all these numbers, {/j. — n){s—l) is 
certainly positive ; so that the first s — 1 terms in our expression 



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266 NOEMAL INTEGEALS [86. 

are not affected by the quantities coiTesponding to -a^i^ + im — )j.)s, 

and they can occur only through the quantities corresponding to 

w^ + (m - X) s, 

for X= 0, 1, ...,n, where T>r„ = 0, and n is the characteristic index 
of the original equation. We thus consider the first s ~ 1 terms in 

and this holds for any value of s equal to or greater than two. 
As regards g, which is the greatest among the quantities 

it occurs only among the first n, in the present circumstances ; for 
it certainly is greater than unity and if any one of the last m — n, 

(say — liT^ is the greatest of these last m — n), is greater than unity, 

then because 

we have 

n /J- \li /\n 1 

that is, 



for /i is greater than n. Thus g does not occur in the last m — ii 
of the quantities, if one or more than one of them is greater than 
unity ; and it certainly does not occur among them, if no one of 
them is greater than unity. Hence g is the greatest among the 
quantities 

It may occur several times in this set ; let - tct, be the first occur- 
rence, in passing from left to right, and - ^^ be the last. Take 
first s = g\ then we have 

■^^■\-{wi, — k) h = mg, OT,. •\-{rn, — r)s = vig, 
■wx + (m - \) s < mg, if \ < k, or if X > r ; 



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8(5,] THEIR DETERMINING FACTOR 267 

SO that the highest terms of ail, being those with index Trig, 
occur in 

ii'", p,ii'"-~ + ... +prn.'«-': 

K then 

p, = x''^-{c, + d„a; + ...). (,7 = 1, 2, ...), 

the equation which determines a^, the coefficient of a:~i' in il', is 

o/ + c^a/-' + . . . + c,. = 0. 
The remaining g — 2 coefficients in il' are given by equating to 
zero the coefficients of the next ^ — 2 terms in 

fl'" + piil'"-' + . . . + p„_iii' -I- p„. 

Each set of values of the coefficients determines a possible form 
of Xi' and therefore a possible form of determining factor. The 
number of sets, different from one another, is ^ r. 

The preceding cases arise through s = g; but if g, being an 
integer, is greater than 2, other values of s, less than g, may be 
admissible. They can be selected as follows*. Mark the points 
0, n; -a-,, n-l; wj, n - 2 ; ...; «r„, 0; 

in a plane referred to two rectangular axes; and taking a line 
through the first of them parallel to the axis of a;, make it swing 
round that point in a clockwise direction, until it meets one or 
more of the other points ; then make it swing in the same direc- 
tion round the last of these, until it meets one or more of the 
remaining points ; and so on, until the line passes through the 
last of the points. There thus will be obtained a broken Hue, 
outside which none of the marked points can lie. 

If a line be drawn through any of the points, say sj,, n — v, at 
an inclination tan"';t to the negative direction of the axis of ^, its 
distance from the origin is 

(i+^r'K+(»-«>*'l, 

so that, for a given direction /a, the distance is proportional to 

It therefore follows that an appropriate value of s is given by any 
portion of the broken line, which is inclined at an angle tan~"'/i to 

' Thy method is due to Fuiseux; see T. K, S 96. 



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NORMAL INTEGRALS 



the negative direction of the axis of y, where ^ i 
integer, ^ 2 : the value of s being 



As many values of s are admissible as there are portions of the 
broken line with inclinations tan~^/(., where /4 is a positive integer, 
which is > 2. 

For each admissible value of s, arising from a portion of the 
broken hne, the terms in 

which correspond to the points on that portion, give the tertns of 
highest negative power in x. If, for instance, a portion of line, 
having as its extremities the points corresponding to 

p,Q'"~' and p(il'"~', {t>r), 

gives a value g' (necessai-ily an integer, as being a value of s), then 
the coefficient a^- satisfies an equation 

Cra/~'' + ... + c, = 0, 

and the remaining ^ — 2 coefficients in li' are obtained in the 
same manner as before. Each set of values of the coefficients 
determines a form of li' and therefore also a possible determining 
factor; and the number of sets different from one another is 
^t — r. 

And so on, with each piece of broken line that provides an 
admissible value of s. 

III. When the greatest of the quantities 



is greater than 2 but is not an integer, we construct a tableau of 
points as in the preceding case, and draw the corresponding line. 
Only such values of s (if any) are admissible as arise from portions 
of the line, which are inclined at an angle tan~^ p. to the negative 
part of the axis of y, fj. being an integer > 2, 

87. In every case, where a possible form of il' and thence a 
possible form of fl have been obtained, we take 



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87.] SUBNORMAL INTEGRALS 269 

If a normal integral of the original equation exists, the equation 
for M must possess a regular integral ; and each regular integral of 
the latter determines a normal integral of the former having the 
determining factor e". An upper limit to the number of integrals 
thus obtainable is furnished by the degree of the indicial function 
of w ; but the investigations of the last chapter shew that, when 
the degree of the indicial function is less than the order of the 
differential equation, the number of regular integrals may be less 
than the degree and might indeed be zero. The simplest mode of 
settling the matter is to take a series of the appropriate form, 
determined by the indicial function of the w-equation, substitute 
it in the differential equation, and decide whether the coefficients 
thence determined make the series converge. The normal 
integral exists or is illusory, according as the series convei-ges 
or diverges. 

When the normal integral exists, we say that it is of grade 
equal to the degree of il as a polynomial in ar^. 



Subnormal Integrals. 

88. In the preceding investigation of normal integrals, it was 
essential that the number s should be an integer ^ 2 : and 
accordingly, such values of j/., as were given by the Puiseux 
diagram and did not satisfy the condition, were rejected. But 
though they are ineligible for the construction of normal integrals, 
they may be subsidiary to the construction of other integrals. 

Let /t denote such a quantity, given by the Puiseux diagram 
in the form of a positive magnitude that is not an integer: its 
source in the diagram makes it a rational fraction which, being 
expressed in its lowest terms, may be denoted by h-i-k. The 
terms which, for this quantity as representing a possible degree 
for il', have the highest index of (ir' in 

Il'« + piil'"-^' + ... +p„-,.ri' +p,„ 
are those which corr^pond to points on the portion of the line 
that gives the value of /t. Hence, taking 



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270 SUBNORMAL [88. 

an equation is obtained by making the aggregate coefficient of 
this term of highest order disappear ; the equation determines A. 
Now take a new independent variable f such that 

and make it the independent variable for the differential equation ; 

dD. A 
ao that 

and therefore 

h — k ^ 

Thus ii is infinite when a: = 0, provided h>k, that is, for values of 
jj. that are greater than unity. Accordingly, when we proceed to 
consider the differential equation with ^ as the variable, values of 
fi of the preceding form can be obtained by the earlier method : 
in fact, we may obtain a normal integral of the equation in its 
new form, the conditions being that the equation for v. which 
results from the substitution 

shall have a regular integral or regular integrals. When once the 
value of k is known and the transformation from a; to |^ has been 
effected, the remainder of the investigation is the same as for the 
construction of normal integrals of the untransforraed equation. 
Examples will be given later, shewing that such integrals do 

exist. As they are of a normal type in a variable x", where k is a. 
positive integer, they may be called subnormal'^. Their existence 
appears to have been indicated first by Fabryf, 

89. We have seen that, if g denote the greatest of the 
quantities 

^x, W^. \^,, ..., 

* Poiiicare, Acta Math., i, viii, p. 304, ealis them anormaUs. 
+ Sar Us intigrahs du iquations di^'erentieiiea linSaires b, coefficients rationels, 
jThfise, 1883, Gauthier-ViUttra, Paris), Section iv. 



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89.] INTEGRALS 271 

and if the equation possesses a normal or a subnormal integral of 
the form 

then il' is a polynomial in ^~' (or in some root of 3~') of order 
equal to or less than ff ; and therefore il is a polynomial in s~^ of 
order equal to or less than g — 1- Let 

then Jt is called the rank of the differential equation for s—0. 

When li is an integer, the grade of a normal integral may be 
equal to M: if not, it is less than B. When R is not an integer, 
let p denote the integer immediately less than R ; the grade of a 
normal integral may be equal to p or may be less than p. When 

k 
R Ts 3, fraction, equal to -: when in its lowest terms, then a sub- 
normal integral may exist having a determining factor e", where 
fl is a polynomial of degree k in z ' ; it will still be said to be of 
grade -j in s, that is, of grade R. All subnormal integrals are of 
grade R or of grade less than R. 
Ex. Olitaiii t)ie rank of the equation 



for 2 ^ CO , the coefB.oiBiits p being polynomials in £. 

90. The converse proposition, due^ to Poincare, is true as 
follows : — 

If n normal or subnormal functions are of grade equal to or 
less than B, and have the origin for an essential singularity, they 
satisfy a linear differential equation of order n and rank not 
greater tha R fa 

Any n f net ns "^at sfj ■i 1 r I tt rentiai equation of order 
n : in the j sent case let tie 

^+P ^ ■+ +PnVJ = 0. 

a a 

* Ada Ml 8 p 30 ii s been somewhat altered, so as 

to admit the □ ma and n a tegrals together. 



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POINCARfi'S THEOREM ON 



[90. 



Let the normal and the subnormal functions be arranged in a 
sequence of descending grade: when so airanged, let them be 

so that, if S,, i^a, ..., Rnhe their respective grades, 
R ^ ill ^ -Ra ^ ■ ■ ■ ^ -Rn— ! ^ -Rji— 1 ^ -B«- 

a the fundamental determinant of the n functions, viz. 



Now 









d"-' 



and A„,y is obtained from A by substituting the derivatives of 
order n for the derivatives of order Ji — ?• in the rth row. The 
value of P,. is 

In order to obtain the degree of s = as an infinity of P,., it will 
be sufficient to consider only the governing terms in A and A,(^r; 
and the degree is determined through the differences between the 
two sets of most important terms in the rth rows. Now if 






d'iwp 



= 0. We taku out of 



where 0g_j, is finite (but not zero) when i 
the pth column a fector 

for each of the n values of y; we take out of the mth row a 
factor 

for each of the n values of m ; and then every constituent in the 
surviving determinants A an<i A„,r is finite. The initial terms in 



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90.] NORMAL AND SUBNOKMAL INTEGRALS 273 

these constituents are the same for all the rows except the 
(r — l)th: the difference there is that ai", Oa", ..., a„" pccur in A'„,ri 
while ai""', Oj""^, ..., ««""*■ occur in A', where A'„_, and A' are the 
modified determinants, and Oi, Og, ..., a„ are the coefficients of the 
governing terms in fi,, fl^. -■-^ ^n- Accordingly, if 

A' =As^+..., 
then 

A'„,r = ^V4-..., 

where the other indices are higher than $, and A, A' are constants; 
and therefore 

i ^ 3-inin-ii (Bi+ii eSOp a^iTp ^.^ 

the summation in the exponents heing for values I, 2 n of p. 

Hence 

Now A, being the fundamental determinant, does not vanish 
identically : and as ^ = is an essential singularity, and not 
merely an apparent singularity, A does not vanish when s = 0; 
thus A is not Jiero. It might happen that ^' = 0; but in any 
case, if -ST, denote the order of 2 = as a pole of the coefficient P,, 

we have i^,.Kr(Ri+l}. Thus the largest of the numbers -z^r 

is <.R,+ 1 ^ R + l ; and therefore, for z = 0, the rank of the 
equation < R, which proves the proposition. 

When all the integrals are normal, which is the circumstance 
contemplated by Poincare, the quantities R are integers and the 
determinants A', A'„^, are uniform ; so that the coefficients P then 
are uniform functions of z. The coefficients P are uniform also 
when the aggregate of subnormal integrals is retained : the proof 
of this statement is left as an e 



Note. An equation, which has a number of normal integrals, 
is reducible; so also is an equation, which has a number of sub- 
normal integrals. 

By the preceding proposition, the aggregate of the normal 
integrals (or of the subnormal integrals) satisfies a linear equa- 
tion with uniform coefficients, say JV = 0, of which they are a 
F. IV. 18 



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27i EXAMPLES OF [90. 

fundamental system. Denoting the original equation by P = 0, we 
can prove, exactly as in § 75, that P can be expressed in the form 

where Q is an appropriate differential operator. In other words, 
P is reducible. 

The investigation of the detailed conditions, imposed upon 
the form of P by the possibility of such reducibility, will not 
be attempted here. 

Further, it must not be assumed (and it is not the fact) that 
retlucible equations ai'e limited to equations, which have regular, 
or normal, or subnormal integrals. 

Ex. 1. Consider the equation 

whore p, q, T are holomorphic functions of a: that do not vanish when ^ = 0. 

To investigate the possible kinds of determining factor, we form the 
tableau of points 

0, 3 ; 3, 2 ; 5, 1 ; 7, ; 

and then eonstruGt the broken line. There are two pieces ; one gives ;< = 3, 
the other fi = 3 ; the former joins the first two points ; the last three lie on 
the latter. The possible ospreasiona for O' are therefore 

where a and 3 ^''S uniquely determinate, and y is the root of a quadratic 
equation. 

Of course, the actual existence of normal integrals depends upon the 
actual forms of p, q, r. 

Ex. 2. Shew that the equation 

y"'+^?'y"+^ »'+^ '3'='' 

where p, g, r are holomorphic functions of ,r that do not vanish when si=Qi, 
possesses no normal integrals in the vicinity of j; = G: but that it may 
possess subnormal integrals. 
Ex. 3. Consider the equation 

which has no regular integral, because the indicial function is a constant. 
The numbers s'j, org, ctj are 1, 2, 6; so that j — 2, and we therefore take 
<=3, so that 



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90.] NORMAL INTEGRALS 275 

We have to make the single (s-l) highest power of a.'~' vanish, in the 
eipansion of 

in ascending powera of x ; hence 
so that a is a cube root of unity, and 

Q = ". 
Accordingly, we write 

after reduction, the equation aatisfied by u is found to be 

,„ _ 3M"(a-^+6aa^) 

The indicial equation for x=0 is 

a'(8-l).0, 
which has a single root 3 = 1 ; so that the w-equation possibly may possess a 
single regular integral which, if it esiats, will belong to the exponent 1, and 
so will bo of the form 

As a matter of fact, the w-equation is satisfied by 

as may easily be verified ; and thus the original equation possesses a normal 
integral 

where a is a cube-root of unity. But a may be any one of the three cube 
roots of unity ; and therefore the original equation in y possesses the three 
normal integrals 

.^(^+«^), ^i^+a^^^), e^(x+a:,), 
where <i is now an imaginary cube-root of unity. 

The singularities of the equation given by l+6i-^ = are only apparent 
(§45). 

jEe. 4. Prove that the equation 

has, in the vicinity of x^a^, two linearly independent normal integrals, 
provided a. is of the form pip + 1), where ^ is an integer ^0 ; and obtain 

18—2 



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276 NORMAL INTBGEAIS [90. 

Bx. 5. Prove that each of the equations 

aY' + ars/ - («s + 2*2) 3^ = 0, 
has, in the vicinity of ;i;=0, two hoearly independent normal integrals ; and 
obtain them. 

Ex. 6. Prove that the equation 

has, in the vicinity of ^ = co, three linearly independent normal integrals; 
and obtain them. 

Ex. 7. Prove that the equation 

possesses one normal integral in the vicinity of x = ; and that one normal 
integral is illusory in that vicinity. 

Ea:. 8. Shew that the equation 

(x + %)3fiy"'-^{x'-i-Zx-'i)^'-{a:-\-%)xy.--{1ix'^-5x-^)y==Q 
posaeaaes three normal integrals in the vicinity of x=Q. 

[They are iee" , xe " , xe "loga^.] 

Ex. 9. Prove that a solution of the equation 

is expressed by 

where 

n' = a^-ib, m(X + l) = a(,7 + l)^c. 

(Math.,Trip., Part i, 1896.) 



Hamburgek's Equations. 

91. The conditions, sufficient to secure that an equation, of 
order in and not of the Fuchsian type, shall have a regular 
integral, have not been set out in completely explicit form 
(§1 78, 79) ; and consequently, the conditions sufBcient to secure 
that such an equation shall have a normal integral have not been 
set out in explicit form. The foregoing examples (§ 90) afford 



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91.] hambueger's equations 277 

illustrations of the detailed process of settling such questions in 
individual instances ; and the following investigation* gives the 
appropriate tests for a particular class of equations, which afford 
an illustration of the general method of proceeding. 

We consider the equation 

„ _ a + Ss + j^ 



1 which a must be different from zero (§ 86) if the equation is to 
s a norma! integral. For any integral that occurs, 3 = is 
singularity. For large values of ^, the integrals are 
regular ; and a fundamental system for s= co is composed of two 
regular integrals, which belong to exponents — p, and — pa arising 
as roots of the quadratic equation 

p{p-\} = y. 

These two regular integrals may be denoted by 

where Pj, P^ are converging power-series. As the origin is the 
only other singularity of the equation (and it is an essential 
singularity), it follows that P, and Pj have a = for an essential 
singularity ; all other points in the plane are ordinary points for 
P, and P^. 

The expression of a uniform function having only a single 
essential singularity, say the origin^.aud no accidental singularity, 
is known by Weierstrass's theoremf to be of the form 



where P (- J is a uniform function having all the zeros of the 

original function (the simplest form of P being admissible), and 

ff ( -] is a holomorphic function of - which is finite everywhere 

except at 2 = 0. 

* It ia due to Harabui^er, CrelU, t. an (188S), pp. 238—273. 
+ T. F., § 52. 



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278 SPECIAL EQUATIONS WITH [91. 

The function g may be polynomial or it may be transcendental ; 
the discrimination depends upon the character of the origin as an 
essential singularity for the original function. As the present 
application is directed towards the determination of normal inte- 
grals, the function g ( - J will be taken to be a polynomial in - . 

If the original function has an unlimited number of assigned 
zeros in the plane outside any small circle round the origin, P 

is transcendental. When the number of zeros is limited, P[-) 
is a polynomial in - , which can be taken in the form 

where A; is a finite positive integer, / is a polynomial in s of degree 
not greater than k, its degree being actually k when ^ = oo is not 
a zero. 

The equations to be considered are those which have integrals 

...p,(l), ^p,(i). 

as above, one (or both) of the functions P, and Pj having only a 
limited number of zeros outside any small circle round the origin, 
with the further condition that the essential singularity at the 
origin is of the preceding type. Thus an integral is to be of the 
form 

say, where il is a polynomial in - , the exponent o- is a constant, 

and /(z) is a polynomial in z; and the differential equation for u 
is to have a regular integral which, except as to a factor z", is to 
be a polynomial in z. Let 

then the equation for u is 



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91.] NORMAL INTEGRALS 279 

After the earlier explanations, it is clear that we must take 



The equation for u then is 



" 2a , , 2a- 0- ye 



--0, 



which is to have a regular integral of the type 
u = .'/(,) 
= z''(c„ + c,z+ ■■■ + C„3" + ...), 
there being only a limited number of terms on the right-hand 
side. The indicial equation for s = is 

- 200- + 2a - /3 = 0, 
BO that 

■2a 
Substituting the expression for u, and equating coefficients, we 
have, after a slight reduction, 

{(n-\-a)(n + ^-l)^y]c, = {2a(n + <r} + ^]c^+, 
= 2a(n+l)c„+,; 
and therefore 

(n + a) (n + <r-l)-'y 
'"■'+' 2«(h+1) 

It is clear that, if the series with the coefficients c were to be an 
infinite series, it would diverge and the integral would be illusory. 
For this reason also, as well as by the initial condition, all the 
coefficients from and after some definite one, say after Ck, must 
vanish ; and therefore we must have 

or substituting for er its value, we see that the qi.iadratic equation 

whff)-e o? = a, m.iist have a positive integer {or zero) for a root. 
This condition is sufficient to secure the significance of the series, 
and therefore sutScient to secure the existence of a normal integral 
of the equation 

" _ « + /3 s + "/g' 



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280 A CLASS OF EQUATIONS [91. 

Clearly, there are two values of a. If for either value the 
condition ia satisfied, there is a normal integral of the form 



where a has .the value for which the condition is satisfied. 

The condition cannot be satisfied for both values, if the values 
of <r are different, and arise from different values of p ; for if it 
could, we should have 

2a 2a 

Now pi + /3i= 1 ; and therefore 

il + 7^2 = pi - 0-, + p, - (Tj = - 1, 

which is impossible, as neither ky nor k^ is negative. 

The condition can be satisfied for both values of a, if the 

values of (7 are the same, that is, if 

13 = 0: 

for then the condition, that the equation {0 + l)O=j can have a 

positive integer as a root, shews that the equation 

„_ a^+ 7^ 
w -- ^ w 

possesses two normal integrals of the form 

e^aCCo + ca+.-.+c^"), 

e"'s(c,-c,z+...±ce^''). 

The condition can be satisfied for both vahies of a, if the 
values of it arise through the same value of p, whether they are 
the same or not ; and the equation then possesses two normal 
integrals. The limitations on the constants are given in the 
first of the succeeding examples. 

Ew. 1. Prove that the equation 

w s4 ■"■ 

possesses two normal integrals, if 

where q is any integer, positive, u^ative, or zero, and. /i is an integer that 
may not vanish. (Hamhurger.) 



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91.] HAVING NORMAL INTEGEALS 281 

Ex. 2. Obtain the conditions sufficient to secure that the equation 



v/' + 2-^vf + 



[ + 3j + y«M^S^(Z* 



may have a normal integral of the foregoing type. Can it have two normal 
integrals ? 

Eie. 3. Prove that the equation 

possesses two normal integrals, if a is an integer (positive, negative, or zero). 
Ex. 4. Prove that the equation 

^ ■ jfl 

possesses a normal integral if the quadratic equation 

has a positive integer (or zero) for one of its roots for either value of ^a. 
What happens (i) when both its roots are integers for the same value of ^a, 
(ii) when, for each value of ^a, the equation has a positive integer for a root ? 

Ex. 5. Prove that the equation 

V"'-2ft(n + l)^ + 4™(w + l)^4-|^w(™ + l)(™+3)(»i-2) + <i'}«'=0, 

where re is an integer and a, is any constant, has four normal integrals of the 

where ^ ( - ) is a polynomial in - . (Halphen.) 

92. In an earlier paper, Cayley* had proceeded in a different 
manner. If 

where {z) is a holomorphic function of z not vanishing with z, 
we have 

W 3 0(S) 
• CreiJs, t. c (18S7), pp. 286—295; Coll. Mn.l\. Paperf, vol. sii. pp, 444—152. 



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where R(z) is a hotomorphic function of 2 in the vicinity of the 
origin. Further, if 

where (s) is a holomorphic fiinctiou of s not vanishing with s, 
and li is a polynomial in - , we have 



W S (s) 



fJiw, 



say, where Ii(z) is holomorphic in the vicinity of the origin. 
Cayley transformed the equation by the substitution 

and then proceeded to obtain, from the differential equation for y, 
an expansion in ascending powers of 2. When once a significant 
expression for y has been obtained, the value of w can immediately 
be deduced. 

Applying this method to the equation 

„ ^ a + /3g + 7^ ° 
w ^ - w, 

the equation for y is at once found to be 

Hamburger's investigation shews that the integrals of the equa- 
tion in vj are 

„..,»P,(lj, „,.,»p,(l), 

which are valid over the whole plane but have ^ = for an essen- 
tial singularity. If an integral, say w^, has an unlimited number 
of zeros, the origin being its only essential singularity, then* 
any circle round the origin, however small, contains an unlimited 

* I', f.. g§ 32, 33. 



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92.] METHOD 283 

number of these zeros: so that if, in the vicinity of the origin, 
the expression of w, is 

^ (a) would have an unlimited number of zeros within the small 
circle so drawn. The expression for ^ is 

but the function ,) ! has an unlimited number of poles in the 

immediate vicinity of the origin, and so the right-hand side 
cannot be changed into an expression of the form 

where m is a finite integer. Accordingly, the assumed expansion 
is not valid in this case : and the method does not lead to signifi- 
cant results. 

But when the integral has only a limited number of zeros, so 
that ^{z) is expressible in the form 



in the vicinity of s = 0, where ^ f - j is a polynomial in - and /(s) 

is a polynomial in a that doe; 
be changed into an expansion 



is a polynomial in a that does not vanish with z, then "^ can 



and so the assumed expansion for p is valid in this ease. The 
method therefore does then lead to a significant result*. 

Assuming the method applicable, and returning to the equa- 
tion 

* The disorimination between the cuses, and the explanation, are due to 
Hambni^er, Crdle, t. cm (1888), p. 342. 



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284 

we easily find 



SUBNORMAL 



2ajao + Uj' — a, = y, 

and, for any value n which is greater than 2, 

2(a„<i„ + «^.ai+...} + (jJ-3)a„_, = 0. 

If the constants in the equation were unconditioned, the co- 
efficients thus determined would give a diverging series for y. 
But we are assuming that the method is applicable, so that the 
conditions for convergence are to be satisfied ; and then, as 



.-(c+a 



■■), 



where the last series converges. The method does not, however, 
give the tests for convergence of the series for y, at least without 
elaborate calculation : still less does it indicate that the con- 
vergence of the series for y is bound up with the polynomial 
character of the series in the expression for w. It can therefore 
be regarded only as a descriptive method, capable of partly 
indicating the form of integral when such an integral exists : 
manifestly, it is not so effective as Hamburger's. 

But the method, if thus limited in utility, has the advantage 
of indicating an entirely different kind of integrals of the original 
differential equation, which are in fact subnormal integrals, though 
it does not establish the existence of such integrals : for the latter 
purpose, other processes are necessary. It will be sufficient to 
consider an equation, say of the fourth order, in the form 



where the origin is a pole of p^ c 
Taking 



multiplicity ct^, for ^ = 1,2, 3, i. 



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92.] INTEGRALS 

we have 



— = y +^yy +y^ 

— = y'" + ^yy" + ^y"' + ^'y + .'/'. 

so that the equation for 1/ is 

if + i^f + Sy" + Gfy +y' + p, (y" + 3i/y' + f) 

+ Pi iy' + y^) + Pay + P4 = 0. 

If this equation is satisfied by aii expression of the form 
y = z--^ {a, + a,^ + ...), 

the coefficient of the lowest power of z must vanish. Now the 
governing exponents for the terms io succession are 

-TO- 3, -2m -2, -2m -2, - 3m - 1, - 4to, 

— wi — m — 2, — BTi — 2m — \, — cti — 3to, 

— OTs — m — I, — ^^ — 2m, 



To determine which groupings of terms will give the lowest power 
of z, we use a Puiseux diagram*; and in connection with each 
quantity ra-^ + km + 1, for the various values of fi, k, I, mark a point 
{ot^ + 1, k) referred to two rectangular axes Ote, Oy. Through the 
point (0, 4) take a line parallel to the axis Ox, and make it swing 
in a clockwise sense until it meets one or more of the points : 
round the last of the points then lying in its direction, make it 
continue to swing until it meets some other point or points ; and 
so on, until it passes through the point (in-,, 0). A broken line 
is thus obtained ; the inclination of any portion to the negative 
direction of the axis Oy being tan~^/i, the quantity ^n is a possible 
value of TO, and the terms giving rise to the lowest index of s in 
the differential equation for y are those which correspond to the 
points on that portion of the line. There are as many possible 
values of m thus suggested as there are portions of the line. 



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286 EXAMPLES OF [92. 

It is not, however, a necessity of a Puiaeux diagram that only 
integer values of m. shall thus be provided : and it does, in fact, 
frequently happen that rational fractional values arise. Let such 

an one be - , where r and s are prime to each other ; and take 

When the independent variable is changed from s to %, an expres- 
sion for y <ii this type can be constructed, and it will be a formal 
solution of the equation ; if the series for y converges, then such 
an integral exists, expressed in the form of a series of fractional 
powers, and a corresponding integral w will be deducible. Such 
an integral, when it exists, is a subnormal integral. 

It is easy to verify that the only points, which need be marked 
in the diagram for the purpose of obtaining the possible values of 
m, are those which correspond with the quantities 

4m, «r, + 3m, to-, + 2m, OTj + m, ra-,, 
as in § 86 ; but fractional values of m are now admissible in every 
case, instead of being so only under conditions as in the former 
use of the diagram, 

Ex. 1. This indication of integrals in a aeries of fractional powers was 
applied by Caylej and Hamburger, in the memoirs already cited, to the 



-(S-S)- 



which possesses neither a regular integral nor a normal integi'al in 
vicinity of 3=0. 

The only points to be marked for the Puiseus diagram are 0, 3 ; 3, 
there is one portion of line, and it gives 

Accordingly, we take 

and the oquation for w then becomea 

^_ldw_ C^.^Cx 

or, wr.tmg 

w = f* W, 

" This equation ie used only for purposes of illustration; its integrals 
regular in the vicinity of s — <n , 



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92,] SUBNORMAL INTEGRALS 

dHV 






ivhich is a special fonii of the earlier equatiou in § 91. It possesses two 
integrals, normal in (, if the quadratic 

(l(e + l) = 4y' + f 
hai> one of its roots an iiit«ger, that is, if 

y = TV(25-l)(25 + 3), 
where 6 is any positive integer {or aero). 

To tied the integrals, we have merely to adapt the solution in § 91, by 
taking 

a = 4p; ^ = 0, v = V + f = S(^ + l)- 
Thus ra = (.* = 3(3'*, o- = l, and 

^(n-e){n + + 1)0^; 
and so, taking C5 = l, we have 

as a normal integral of the equation in f. Accordingly, the equation 



whore e is e 


I positi' 


ve integer or 


zero, and a ii 


3 a constant, has 




. 


=.^i'"V 


i-K-iiJ 


' {6+n)] 


.4- 


Manifestly, 


the other integral i 


s ^ven by 








„ 


,^,-u.-s^ 


'im 




4, 



the two constituting a fundamental system, Eaeh of them is of the type of 
normal integral ; but the aeries proceed in fractional powers of the variable. 

It will bo noted that the two values of <r Bxe the same, and that only 
one yalue of p is used ; the relation is 



Sj:. a. Prove that the equation 

where X is a constant and 2fi is an odd int^er, pc 
two subnormal integrals. 



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EQUATIONS HAVING 



Equations op Higher Order having Normal or 
Subnormal Integrals. 

93. There is manifestly no reason why Hamburger's method 
should he restricted to equations of the second order ; and he has 
applied it to obtain the corresponding class of equations of general 
order, the properties of the integrals defining the class being 

(i) the integrals are of the regular type in the domain of 

(ii) the origin is an essential singularity for each of the 
integrals, and at le^t ono of the integrals must be of 
the norma.l type in the vicinity of s = ; 

(iii) all the points, except z = and z = oo, are ordinary 
points of the integrals and the equation ; 

(iv) the number of zeros of at least one integral, which lie 
outside any small circle round the origin, is limited ; 
the second and the fourth of which are not entirely independent. 

Let the equation he of order n, and have its coefficients 
rational. The first of this set of properties requires the equation 
to be of the form 



where Pups, ...,pn are holomorphic functions of s for large values 
of z, and thus are expressible in series of powers of 3 of the form 

a^ + h^- + G^-^+ — (/i=l, ■■■,n). 

The third of the above set of properties requires that every value 
of z, except s = 0, shall be an ordinary point for each of the 
coefficients : and hy the second of the properties, s = is a singu- 
larity of the equation and therefore of some of the coefficients. 
Accordingly, the power-sfcries for the coefficients p, which have 
been taken to he rational and are limited so that every point 
except 2^ = is ordinary for them, are polynomials in zr^. 



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93.] NORMAL OB SUBNORMAL INTEGRALS 289 

As the integrals are regular in the vicinity of s = a> , one at 
least is of the form 



where Q is a series of powers of z~', which does not vanish when 
z=<x: and converges for all values of z outside an infinitesimal 
circle round the origin, and where p is a root of the equation 

p(p-l)...{p^n + -i) + a,pip-l)...(p-n + 2) + ... 

the indicial equation for 2=x. The exponents to which the 
integrals belong, being regular in the vicinity of z= 'x> , are the 
roots of this equation with their signs changed ; and they exist in 
groups or are isolated, according to the character of the roots. 
Let the above integral be one which, under the second of the set 
of properties, is a normal integral in the vicinity of s = 0, neces- 
sarily an essential singularity; in that vicinity, its expression is 
of the type 

where R {z) is a function of z, which is holomorphic in the vicinity 
of 2 = and does not vanish when s = 0, and where fl is a poly- 
nomial in «~^, say 



Ii = 



* - 1 3'«-i 



and (7 is a constant, Then, in the vicinity of ^ = 0, we have 

VJ z^+i s™ z^ s R (z) 

= r + B„ 

where T is a polynomial in - , constituted by U' -|- - , and i£i ie 

the holomorphic function of s given by R' {z} -r R (s). But as 
this arises through a form of the integral, postulated for the 
vicinity of z = 0, while the integral is actually known to be 



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the above form for v>'jvj must be deducible from this actual value. 

This is possible only if (^[-)i which has s = for an essential 

singularity, possesses at the utmost a limited number of aeros 
outside an infinitesimal circle round the origin ; for if it had an 
unlimited number of zeros in the plane, other than ^ = 0, any 
circle round the origin, however small, would include an infinite 
number, and then 

"<} 

would be incapable of such an expansion. The requirement, that 
thus arises, has been anticipated by the assignment of the fourth 
among the set of properties of the integrals; and so we may 

assume Q{-\ to have only a limited number of zeros. Accord- 
ingly, as in § 91, the form of 6 { - ) must be 

where -P(-) is a polynomial in - having as its roots all the zeros 

of Q (- j , and g[-\ is a holomorphic function of ~ , tinite every- 
where except at z=^. 

Let k be the number of zeros of Q ; then P f-j is a polynomial 
of degree k, an<l so it can be represented in the form 

where {z) is a polynomial in z of degree k. Thus the integral is 
of the form 

The postulated form must agree with this form; hence (I [-\ 

is the polynomial li of that form, and the holomorphic function 
R{p) of that form is the polynomial G{z) : also 



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93.] INTEGRALS 291 

The expression for w'/w in ascending powers of z is thus valid, 
under the conditions assigned, provided ii(s) is a polynomial in z. 
Taking 

so that Pi is a function of z, which is holomorphic in the vicinity 
of 3 = and is equal to a™ when 2 = 0, we have 

Then 

10 \w) dz\w} 

= Z-^-^ {P,= + 2^Qx) = ^-''"T'P,, 

say. Similarly, 

"w = ^~"''" ^ ^'' '•' ^"' ^'* " ^"""""-Ps, 

say, and so on: where all the functions Pg, P3, ..., Qj, Qj, ... are 
holomorphic functions of z, and the first m terms in P, ai'ise from 
Pi". Substituting in the equation 

d'^w d'^^w dw 

^2" -^ d3"~^ ' dz ' 

we have 

P„ + z'^ih Pn-l + 2'>=-P»-s + ■ . ■ + 2"™p« - 0, 
which must be identically satisfied. The coefficients p are poly- 
nomials in -: hence* 
z 

z-^p, 

is expressible as a polynomial in z, and so the highest negative 
power in p^ is z~'™ at the utmost. Accordingly, let 

J- ^ z z^ z'-^ 



Now we have 

F^ = rt,„-\- «™_,2 + ... + MiS""-' + ^2"= v + ^2", 

•^ If this were not the case, the aesignment of a larger value of m could Bet 
it: SDd GO the assumption reailj is no limitation beyond that wliicli is neoesBarj 
a noiTEial integral, viz. la must be a finite integer. 

19—2 



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292 CONSTRUCTION OF [93. 

say, where T is a holomorphic function of z ; and 

where T^-x is a holomorphic function of s ; so that the first m 
terms in P,, which give all the coefficients in the exponent of the 
determining factor e", are given as the first in terms of a root of 
the equation 

u" + z™pi ii"~' + s™^5 !!"-= + . . . + Z^^'pn = 0, 

when the root is expanded in ascending powers of z. When the 
first «i terms in v are obtained, then the determining fa<;tor is 
known ; for we have 



li= {' ar"^-^v 



Moreover, after this determination, the terms involving the powers 
^, z\ ..., ^"^' in 

have disappeared, so that this quantity is divisible by z™, leaving 
a holomorphic function of s as the quotient. 

94. Having obtained the determining factor, let 



be substituted in the differential equation, which can now be 
taken in the form 

For this purpose, derivatives of e" are required. We have 



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94.] NORMAL INTEGRALS 293 

and so on, where v is identical with the first m terms of Pi, 
Vi is identical with the iirst m terms of P^, and generally, Vx is 
identical with the first m terms of Pk±^. Now 

^ _ e" 2 I - J^ji^^"^ 

with the convention v„ = v, U-i = 1 ; and therefore the equation 
for u, after dropping the factor e^, is 

which can be written in the form 
where j)o= 1- The coefficient of m is 

Because the first to terms in ^a-i are the same as in P^, the first 
TO terras in the preceding coefficient are the same as in 

P™ + ^'"PiPn-i + . . . + ^""Pn, 
and they are known to vanish, for the coefficients of s", s'', ..., 2"'~' 
were made zero to determine v ; hence the preceding coefficient is 
divisible by a™, so that we can take 

if„_, + £"'piV„_^ + ... + z'"^p^ - 2^(0, + e,z + ...), 
where 6^ is a determinate constant, because v is known. 

The coefficient of 2™+^ -^ is 
dz 

= mUn-a + (Jl - 1) IJ,^3™^1 + . . . + Sv^f-" "i)^a + ^f"^^' "'p^i. 
The first m terms here are the same as the first to terms in 

mP„_i + {n- l)z"'p,P^, + ... + aPis'^-^'^^^H- af-^^iJ^-i, 
that is, the same as the first m terms in 

„„«-! +(n~l) v'^-^z'^p, + . . . + 2vz("-^^ "'p^-^ + z '"-" ^'pn-v 



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294 CONDITIONS FOR EXISTENCE [94. 

The equation for v is 

H" + 3™Pj^"-' + z^"'p^v'"^ + ... + z"™p„ = ; 
and, in particufar, the equation determining a^, the constant term 

giving n values of o^, 

95. Let a,„ denote a simple root of this equation, sometimes 
called the characteristic equation : then th^ quantity 

«£»«,''"' + (»~l)«™""'ni,» 4- ... + «„_i,«m-m 

is not zero. The coefficient therefore of z™^' y , as given above, 
does not vanish when s = r let it be 

'7o + '?i2+ -■-. 
where 5jo is a determinate constant, because v is known. 

It follows that the equation for it, in the form as obtained, is 
divisible throughout by s*^. Further, if it possesses (as, for the 
class of equations under consideration, it must possess) a regular 
integral, and if that regular integral belongs to the exponent a, 
then a is given by the' indieial equation 

V„<y + t^o = 0, 

so that u can now he regarded as a known constant. 

Further, we had 

where & is a positive integer (or zero), and p is a root of the 
equation 

p{p-l)...{p-n+X)^p(p-\)...{p-n^2)a,, 

+ p{p-\)...{p-n-\- 3)aai + ... + /5a„_i,o+ano=0, 
say, of 

'(p) = 0. 
Consequently, the equation 



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95,] OF NORMAL INTEGRALS 296 

regarded as an equation in k, must have at least one root equal to a 
positive integer or zero : if this root be denoted by k, one condition 
that u should be of the form 

u = a" {c„ + c^e + . .. + c^if) 
(which is the fonn for u required by the earlier argument) is 



But while the condition is necessary, it is not sufficient for the 
purpose. When the value of u is substituted in the equation, the 
latter must be identically satisfied ; and so we have relations 
among the coefficients c. The general relation is 

/(o- + a) o.+g,(a)c^^,+g,(a)c^, + ... + £r™„^™(a) c,+™^™ = ; 

the relations for the first few coefficients are of a simpler form. 
When these relations are solved, so as to give successively 
the ratios of c,, Cj, ... to Co, a formal expression for u is obtained. 
In this forma! expression, all the coefficients c,+,, c^+2, ... are 
to vanish ; that this may be the case, we must (as in § 79) have 

/(ff + «)c, = 0, 

7 (<r + « - 1) c,_i + 9, (« - 1) c, - 0, 

/(<7 + «-2)c,_ + ^,(«-2)c,_, + 5.(«-2)c, = a 

and so on, being m.(n — 1) relations in all. Of these, the first is 
known to be satisfied as above; it is the first condition for the 
existence of u in the specified form. The aggregate of conditions 
is sufficient, as well as necessary: the last of them secures that 
c,+i vanishes, the last but one secures that c^+j vanishes, and so 
on : the first secures that c»+mn-m vanishes ; and then, in virtue 
of the general difference-relation among the constants c, every 
succeeding coefficient vanishes. 

Thus when the m (ji — 1) conditions arc satisfied, in association 
with a simple root of the equation 

a normal integral of the original equation exists. 

It may happen that the conditions are satisfied for more than 
one of the simple roots of the equation : then there will be a 
corresponding number of normal integrals of the equation. 



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296 NORMAL [95. 

The extreme case would be that in which every root of the 
equation 

is simple and the conditions are satisfied for each of the roots ; 
there then would be n normal integrals. Let the n roots be 
denoted hy 6^, 8^, ..., On, so that, if 

the normal integrals will be of the form 

where Uy is a polynomial, say of degree Xr, in s- We have 
(Tt = p -- Kt ; when these n indices a-,, arc associated with n 
quantities p, it follows that 

for r = l, ...,». The distinct quantities p^ are the roots of 
/(p) = 0, so that, if they are all different from one another, we 
have n of them ; also 

i_(„, + ,,) = i„(,>-l)-a,.. 
The value of 2 o-^ can then be obtained as follows. 
Construct the fundamental determinant 

\ dz ' dz ' ■"' dz 



which is equal to 
that is, to 



rft-. 



where A 1 



1 constant. Now if w^ = 1 + c„j 



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95.] INTE(3RA.LS 297 

where u^, is a polynomial in z which is equal to 1 when s = ; also 



where Un is a polynomial in s which is equal to 1 when e = ; 
and so on. Thus 






„+^„+..-i.,.- 


)(.+i)(i.(^), 


2). 


1 +... 


1 + 






e, +... 


9, + 






«,■ + ... 


«,"+,.., ... 





As the roots are unequal to one another, <I>(z) does not vanish 
when 3 = 0; and it is a polynomial. We thus have 



-*"l''-'"'"+"3><ii)-^«~-( 



Accordingly 



iii + ... +fl„ = — + ... + 



1 Ci.ir 



^S<r,-i»(»~l)(m + l) = -«,, 

that is, "l" (s) reduces to its constant non-vanishing terra. Thus 

i^„,-i»(»-l)(m+l)-o,.. 
We saw that 

I (,r, + «,) = i»(n-l)-«,.; 



which is impossihJe because no one of the integei's k, is negative. 
It therefore follows that when the characteristic equation 



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298 MULTIPLE ROOTS OF [95. 

has all its roots distinct from one another, and when the quantity/ 
denoted by u has n distinct values, associated respectively with 
n distinct roots of I(p) = 0, the differential equation 






where 

cannot have more than n — 1 normal integrals, linearly independent 
of one another. 

If, however, the quantity denoted by a liaa fewer than n 
distinct values, so that it could he the same for more than one 
of the n distinct quantities li, the relation 

l^^. = i«(«-l)(m+l)-a,„ 

would still hold, repetitions occurring on the left-hand side. But 
in that case not all the roots of the equation / (p) = are specified, 
for the same value of le could be associated with the value of 
a- common to two integrals; and the relation 
S(,T + «) = i«(»-l)-a.. 
no longer holds. The theorem then cannot be inferred as neces- 
sarily true : and it will appear from examples that an equation in 
such a case can have a number of normal integrals equal to its 
order. 

Similarly, if o- has n distinct values, and if these values are 
not associated with n distinct roots of / (p) = 0, the preceding 
theorem is not necessarily true; the differential equation can 
have a number of normal integrals equal to its order. 

96. Next, let a„ denote a multiple root of the characteristic 
equation 

a™" + Om""' ffli, im + ■ . ■ + c[»,«™ = ; 
then the quantity ?jo vanishes, where 

lJ, = )iam"-^ + {>l — l)a,^"'^«i,im + ... +an-i,wm-m- 

The indieial equation is 

and cr must be a finite quantity. If 0„ is not zero, the latter 



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96.] THE CHARACTERISTIC EQUATION 299 

condition is not satisfied: and tlien the original equation lias no 
normal integral to be associated with that multiple root. If B^ is 
zero, the preceding indicial equation is evanescent : and so further 
consideration is required. The differential equation for «, on 
division by 3™+', becomes 

dn 





(»,+ 


9,J+. 


+ 2-+> (*. + *, 


wh. 


ere the 


coefficient of ',-^- is of the form 


for 


r = 3, 


4 


2'-+'"'~"(t. + +.^ + 




When 


ni^l, 


the indicia] eqnation is 


when m > 


l.the 


indicial equation is 



In either case, we can have a possible value for cr. A regular 
integral of the equation for ti, and a consequent normal integral 
of the original equation, exist if the appropriate conditions, 
corresponding to those for a simple root, are satisfied : it is 
manifest that they become complicated in their expression*. 

97. It might happen that, in determining v, one or more 
roots of the equation 

«7«" + «!»""' «!, im + ■ ■ ■ ■+ (^i, nm — 

is zero, while some of the remaining coefficients in v do not 
vanish ; the implication is that (other conditions being satislied) 
a normal integral exists, having a determining factor of which 
the exponent is a polynomial with a number of terms less than m. 
It might even happen that, with a zero value of a^, all the associ- 
able values of the rest of the coefficients are zero, so that v = 0, and 
the determining factor disappears. One possibility is the existence 
of a regular integral, and the possibility can be settled in the 
particular case by the method given in Ch. vi. If, however, the 
conditions for a regular integral are not satisfied, then there is the 

' They are considered by Giinther, Cnlle, t. cv (1889), pp. 1—34, in particular, 
pp. lOetseq. 



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SUBNORMAL [97. 

Hty of a subnormal integral of the original equation ; it 
IS follows. 



Lot 




«.=(«,. 






substituted in the equation 








d'Hv 
"dz^ ^ 


!£!-- 


fj»=o; 




en the 


equation for \ 


.,. is (by § 85) 








S+fi 


^--)|S- 


... + g.«. 


■0, 


lere 










d 


?,- 




f,. 





Now fi is to be chosen so as to diminish the multiplicity of a = 
as a pole of g„. After the preceding hypotheses, we shall not 
expect to have an expression of the form 

(-1/ _ Uj , ^ I , _^~''i__ 

where m is an integer; but after the indications in § 92, it is 
possible that II' may be a series of fractional powers. Accord- 
ingly, assume that the multiplicity of s = as an infinity of li' is 
/t, so that ^''fi' is finite when 3 = 0: then in ^n, we have a series 
of terms with infinities of oi-ders 

n/i , (n- l)/i-|- 1 , ... 

(ji-l)^+ m + l, («~2)^ + m + 2, ... 
(?i. -2)^+ 2m + 2, (n-3)y^ + m-|-3, ... 

n{m + \). 
Construct a Puiseux tableau by marking points, referred to two 
axes, and having coordinates 

0, « ; \, n-\\ ... 



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97.] INTEGRALS 301 

(it is easily seen to be necessary to mark only the first in each 
row), and construct the broken line for the tableau, as in § 92. 
If the inclination to the negative direction of the axis of y of any 
portion of the line is tan"' 0, then ^ is a possible value for ft. If 6 
be a positive integer ^ 2, we have a case which has already been 
dealt with. If ^ = 1, there may be a corresponding integral ; but it 
is regular, not normal. If ^ be a negative integer, li' is not infinite 
for 3 = 0, and the value is to be neglected. If ^ be a positive 
quantity but not an integer, it must be greater than unity to be 
effective ; for if it were less than unity, H would not be infinite 
for s = Q. Suppose, then, that 6 has a value greater than unity; 
as it arises out of the Puiseux diagram, it must be commen- 
surable : when in its lowest terms, let it be 



where q and p are integers prime to one another, and q>p. 
Then take 

3 = a* ; 

we have an equation in u and x, and a possible determining factor 
e" can be found such that 

and so 



a series of fractional powers. The investigation of the integral of 
the new equation in u and x, that may exist in connection with 
this quantity fi, is of the same character as the earlier investi- 
gations. 



Equations of the Third Order with Normal or 
Subnormal Integrals. 



98. The preceding general theory, and the methods of > 
with the cases when the equation for a^ has equal roots, or h 
zero roots, may be illustrated by the considei'ation of an equati< 



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302 EQUATIONS OF [98. 

of the third order more clearly than by that of an cqua.tion of the 
second order, as in § 91. Taking the ainiplest value of m, which 
is unity, the equation is of the form 

", ^^lo^ + ^ji '/ k«,sf + kill -^ k-,« , 
w +3 , — w H — — — -; w 

fC^Z -j- iC^S + jCj^Z + fi^aa r. , 

which, on using the substitution 

ti,„z+k„ ^^ iu 

y = we ^ =iuz''">e ', 
becomes 



where the constants a are simple combinationB of the constants k. 
The substitution adopted changes a normal integral of the one 
equation into a normal integral of the other, save for the very 
special case when it might be changed into a regular integral of 
the other: it therefore will be sufficient to discuss the form 
which is devoid of a term in y". 

In the present case, m = 1, we take 

y = e'u, 

and a is chosen so as to make the coefficient of the lowest power 
in the coefficient of w equal to zero. We thus have 

and the equation for u then is 

+ -, [a^e'' + («si - a«™ - 6a) a + (a,, - ««,. - 6^0) = 0, 
of whicli the indicial equation for s :^ is 

It is clear that the equation in a. will not have a triple root : if it 
could, we should have (Xs = 0, «3a = 0, a = 0, the last of which 



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98.] THE THIBD OBDER 303 

values leads to the collapse of the process. (Account must, of 
course, be taken of the possibility that csja = = a,^, and this will 
be done later.) Meanwhile, we assume that a is either a simple 
root or a double root. 

First, let a be a simple root ; then o^a + 3a^ is not zero, and 
the foregoing indicial equation then gives a proper value for a-. 
If — p ia the exponent to which an integral in the vicinity of 
0—x> belongs, p is a root of the equation 

/{p) = p(p-l)(p-2) + a„p + «,, = 0. 

The general investigation has shewn that this must have a root of 
the form p = o- + k, where k is a positive integer (or zero), and 
that, if this condition is satisfied, the form of u is 

M = 3''(C„ + C,S+ ... +0,3"). 

We substitute this value, and compare coefficients. If 
ff„ = {ff + n){ff + n-l)(<7 + ii-2) + a^(a- + n) + a^, 
k„ = -SaX<T+n)(a + n + l)+{a,, + 6o.){^ + n + l) 

+ Ug — aoan — 6a, 

then the difference -equation for the coefficients c is 

for values of m^O, together with 

As a is a simple root of its equation, a^, + 3a^ is not zero : thus all 
the quantities k^,, k^, kj, ... are different from zero, and the pre- 
ceding equations thus determine Ci, C3, ... in succession, say in the 
form 

In order that the integral may not become illusory, the series is to 
be a terminating series : it would otherwise diverge, on account of 
the form of §„■ Let the series contain k + 1 terms ; then all the 
coefficients c»+i, c,+a, ... must vanish. Now c,+i vanishes if 

then c,+a vanishes if 



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304 NORMAL INTEGRALS OF AN [98. 

and then all the succeeding coefficients c vanish. The latter 
condition gives ^,= which, as ^„=/((r+«,) for all values .>■' 
n, is the same as 

/(„ + «).0, 

a known condition ; and the other gives 

which is the new condition. When hoth conditions are satisfied, 
a noiTTial integral exists for the equation in y. As that equati i ■ 
involves seven constants, which are thus subject to two conditio:!., 
there are effectively live constants left arbitrary, subject solely :■■ - 
a condition of inequality as regards the roots of the equation 

moreover, « may be any positive integer (or zero). 

If the corresponding conditions hold for a second simple root 
of this cubic equation, the number of independent constants is 
reduced to three, while there are two integers such as k; the 
differential equation for y then has two normal integrals. 

If all the roots of the cubic equation are simple, and the 
corresponding conditions hold for each of them, there are three 
integers such as k, and there is effectively one arbitrary constant : 
the differential equation for 1/ would then have three noi'mal 
integrals. This, however, is impossible, if there aie three diff<!renfc 
values <r, a', <r" of a, and three associated integers k, k, k", yiich 
that (7 4- K, <t' + ic', <!■" 4- k" are different roots of f{p) = 0. For 
then 



Now we have 






loo? + attji - ^35 


where 


dk 
da 



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98.] EQUATION OF THE THIRD ORDER 

hence, summing for the three roots of h, we have 



by a well known theorem in the theory of equations. We then 
should have the equation 

which is impossible as no one of the integera k, k', k" can be 
negative. Hence, when the equation a? + tto^B — Osj = has three 
distinct roots, and when there are three different values a, a, a" 
of IT, associated with three integers k, k, k", such that <r + «, (r'+ k, 
o-"+ k" are different roots o{f(p) = 0, then the differential equation 



cannot have more than two normal integrals. But, if the values of 
<j- are fewer than three in number, or if the quantities a- + k are 
not different from one another, then the differential equation (the 
other conditions being satisfied) can have three normal integrals. 
Next, let a be a double root of the equation 
ft. = a* + aiXffl — asa = 0, 
BO that we have 

in order that this may be the case, the relation 

must be satisfied. The quantity a-, given by 

(3a^ + £%) (7- + o^y ~ aos, — 6a^ = 0, 
is infinite, unless a^ — aaj, — 6tt' vanishes : if this condition is not 
satisfied, then the regular integral for the w-equation, and conse- 
quently the associated normal integral for the y-equation, cannot 
exist. Hence a further condition for the existence of the normal 
integral is, that the equation 

Ksi — noai — 6a^ = 
be satisfied, where a is the double root ; that is, 



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306 EQUATION OF THE 

Assuming this to be satisfied, the equation for u aow is 

Now 



Sctj, 



21, say; 



so that the equation for u is 

The indicial equation for z — is 
Substituting 

« = 3«(p„+C,3+... +C„3''+.,.) 

in the equation, we have 

Du = Cc^S {C,,f (fl - 1) + C.,ye + Osij, 

provided 

for all values of )i ^ 0, where 

g^=^{e^n)(e^n^l){8 + n-2) + a^{e+n) + a^, 
A„ = c,„(S + m + ])(^ + m) + C^(^ + ?t + l) + c„. 

First, let the roots of the indicial equation be unequal, say X 
and /i, so that 

Du=c,o,„e^{e-\){e- (>.). 

Then the value of u, when 8 — X, gives an expression which 
formally satisfies the equation ; but it has no functional significance 
unless the series converges. That this may happen, g^ must vanish 
for some value of n, say k^, when d ~X; that is, one root of 

I(p) = p(p-l)(p~2) + a^p + a,, = 
must be 

pT=\+ K-i, 



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98.] THIRD OllDER 307 

where «i is a positive integer or zero. If that condition is 
satisfied, then a regular integral of the w-equation and an asso- 
ciated normal integral of the ^/-equation exist. 

Similarly, if l{p)~0 has another root 

p = fi + «:.„ 

where «, is a positive integer, then the value of u, when $ = /j., has 

significance. It is a regular integral of the w-equation; and a 

corresponding normal integral of the original equation then exists. 

Let denote the root of the cubic that is simple : then the 
earlier investigation shews that a corresponding normal integral 
may exist. If a' be the exponent to which the regular w-integrai 
belongs and H k^ + I he the number of terms it contains, then the 
equation /(p) = has a root 

p = a' + >c,. 
But the three normal integrals, each one of which is possible, 
cannot coexist, if X + ati, /j, + k^, ct' + k, are different roots of 
/(p) = 0, supposed not to h&ve equal roots. If they could, we 
should have 

>. + /i + (7 -j- Ki+ IC^ + Ks — ^p — S. 

Now 

X4-w = l- — = 1 - — . 



Also 
and 



for a, a, are the roots of the equation 

a^ + aa^i — agj = ; 



on reduction, after using the value of a and the relation 

fflisa — "«ai — 6a° = 0. 
Hence 

\ + fi + <r'^5, 
and therefore 

/C| -i- /Cs + «3 = ~ 2, 



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308 NORMAL INTEGEAL8 OF AN [98. 

which is impossible, as no one of the integers « can be negative. 
Hence, when the roots of the indieial equation 
C„.r(,r-l) + C„.r + «. = 
are unequal, acd wheo /(p) = has not equal roots, the original 
equation cannot have more than two normal integrals, unless (in 
the preceding notation) there are equalities among the quantities 
X + Ki, /i+Ka, cr'+zcj. If it possesses the two normal integrals 
associated with X and fi, it is easy to see, from the expression for 
kn, that, if \ — /j. be a positive integer, it must be greater than 
Ka + 1 : and that, if /t — X. be a positive integer, it must be greater 
than «i + 1. 

Next, let each of the roots of the indieial equation for <r be 
equal to r : so that 

Thus the two quantities 



."(^-t)^. 



M- [!],..• 



are expressions that formally satisfy the equation : they have no 
significance unless the series converge. That this may happen, g^ 
must vanish for some value of n, say k, when d — r; that is, one 
root of the equation 

I(p) = p{p- 1) (p - 2) + a,„p + «,„ = 
must be p — t + k, 

where «' is a positive integer or zero. (The quantity A„ never 
vanishes in this case and so imposes no condition.) On dropping 
the coefficient Co, the expression for u in general is equal to 



■''/i„A/' ■■■^* ^^AA. 






so that the two integrals are of the form 
V, vlog^ + iJi, 
where v — [u]e=T, and Vi is an expression similar to v with different 
numerical coefficients, viz. the coefficient of (— lyz^'^'' in v, is 

[hX -.*,-, Ui \g. ~tie K dei]\,„ ■ 

The corresponding normal integrals are 

<fv, «■(» log « + ..). 



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98.] EQUATION OF THE THIRD ORDER 309 

A third normal integral can coexist with these two in the present 
case in the form 



where u belongs to the exponent c', = — + 4, provided l{p) = 

has a root of the form a-' + x^, where «, is a positive integer (or 
zero). The reason why three can coexist in this case is that 
only two quantities t and (/ arise, and only two roots, not three 
roots, of I (p) =0 are assigned. 
B^\ 1. Prove that, if the equation 

possesses a, normal integral of tlie form 

the constants j3, n, o- are given by the relations 

(T (3S= + oao) + 3a=S - 9|3^ 4- ("Is, + /3aj2 + ctaa = ; 
and the equation 

p{p-l)(p-3) + p«j4 + <%'=0 
must have one root equal to tr+K, where ic is a positive integer (or zero). 
Obtain the relations sufficient to secure that the series Cg+OjB+.,. shall 
contain only k + 1 terms. 

Assuming that three values of a, distinct from one another, correspond to 
three seta of values of a and ft prove that their sum is 9 : and hence shew 
that, in this case, the differential equation cannot hive more than two normal 

In what circumstences can th i fl t 1 i n i tl 1 

integrals ? 

En). 2. Obtain the constant and tl e nd t ns of e ten e f the 
normal integrals of the equation n the p e ed ng e a nple when a an she? 
and a^ does not vanish. How m ny n mal tgl ntlequtn then 
have? 

99, We now have to c d ( ) tl e ca n 11 ne ze o 
root for a occurs, so that a^ — 0; and (ii), the case in which all the 
roots a are zero, so that a^ = 0,0.^^ — 0. 

Taking a^ = 0, the equation is 

/ + zi — 2/ + —3 — — y=u. 



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310 srEoiAL [99. 

Two non-zero roots arc given by 

a normal integral may exist in connection with each of them. 
The indicial equation for « = is 

in connection with this exponent, a regular integral may exist. 
The investigation of the respective conditions is similar to pre- 
ceding investigations. 

Now substitute in the equation 

the equation for u is 



u"'+3u"n' + u'Uil''' + il"-i 



a^± a^iZ + Oi^X _ _ 

and by proper choice of il, the multiplicity of a = as a pole of u 
is to be diminished. Assume that z~i'-Df is finite {but not zero) 
when z = 0, and form the tableau of points in a Puiseux diagram 
corresponding to 

3|U, 2/4 + 1, iM + t, /1 4- 4, 5, 
that is, insert the points 

0,3; 1,2; 2, 1 ; 4,1; 5,0. 
The broken line consists of two portions : one of them gives /* = 2, 
the other gives fi = \. The former gives the possibility of two 
normal integrals : the latter gives the possibility of one regular 
integral as above. 

But now let a23 = 0, as well as Us^ = 0. The equation for a 
becomes 

a= = 0, 

so that the method gives no normal integral. When we proceed 
to the equation for u, the coefficient of it is 



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99.] CASES 311 

We form the tabloau of points in a Piiiseux diagram corre- 
sponding to 

d/j., V + 1. f^+^' /' + -''. ^: 

that is, we insert the points 

0,3; 1,2; 2, 1 ; 3,1; 5,0. 

There is only a single portion of line ; it gives 

/i = f. 

Accordingly, we change the independent variable by the relation 

s = a^; 
the form of il' is 

of- «* 
that is, 

dll_3a' W'_0 ^ 
rfa; ~ ** "•" a^ "a^'^x'' 

say. The differential equation 



with the substitution y^vx', becomes 



+ ^ {t1a.^ + {tla,, + 180=,) ^ + {^-la^ + 18<i™ + 8) a^j = 0. 
If a determining factor exists, then (Ex. 1, § 98) it is of the form 







^ + 2?o.. = 0, 




that is. 


/3- 


a. 3^= + 9(12,3 = 0, 


- (tjittg 


Substituting 









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[99. 



and using these values of a and /3, we find the equation for u i 
the form 



dx'- 

+ " [- 9;8= + {o? + 63aa + 27a,0 x + |(12a,„ + 4) /3 - 6a^] a^^ 

+ a (9a„ - 2) it^ + (27tt™ + 18a» + 8) a^] = 0. 
If the equation in ^ is to have a normal integral, this equation in 
u must have a regular integral belonging to an exponent cr, where 
it is easy to see that 

The regular integral for -u is of the form 

u = X c,,a:^+\ 
If 

/„ = {n + 3) (« + 2) (n + l) + (9a™ -S){n + 'S) + 21a,, + lSc(,„ + S, 
g^ = 3a(H + 4) {n + 3) - 6a (« + 4) + « (9<i» - 2), 
A„ = 3/3 (■« + 5) (« + 4) + (3a^ - 9,9) (n + 5) + {\2a.^ + 4) ^ - 6a^ 
& = a=+63a.s, + 27aai, 
i„ = 3^^(^ + 4), 
the difference-relation for the coefficients c is 

together with 

fe(, + LiCi = 0, 

A-aCfl + tei + Z-jC; = 0, 
^_iC„ + A_iCi + ACa + Li^s = 0, 

The conditions, necessary and sufficient to ensure that the aeries 
for M terminates with (say) the (« + l)th term, which is the 
generally effective manner of securing the convergence of the 
series, k being some positive integer or zero, are 

four conditions in all. 



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99.] CASES 

Assuming these satis6ed, we have 



M^/i 2 ( 



a subnormal integral. 

If the conditions are satisfied for more than one of the cube 
roots of djg, then there is more than one integral of subnormal type. 
Moreover, the value of ct is the same for all three cube roots, and 
only one value of « is required : so there may be even three sub- 
normal integrals, each containing the same number of fractional 
powers. 

In order that this analysis may lead to effective results, it is 
manifest that «3a should not vanish. 

Hx. 1, Prove that the equation 

possesses three subnormal integrals. 

Ex. 2. Discuss the integrals of the equation 



Normal Integrals of Equations with Rationai. 
Coefficients. 

100. In the discussion at the beginning of this chapter, the 
only requirement exacted from the coefficients was as regards 
their character in the vicinity of the singularity considered: and 
a special limitation was imposed upon them, so as to constitute 
Hamburger's class of equations in §§ 91—99. More generally, 
we may take those equations in which the coefficients are rational 
functions of z, not so restricted that the equations shall be of 
Fuchsian type ; we then have 

Po jiiT -^Pi -JZazT + ■■■ +P»w = 0, 



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314 roiNCARfi ON [100. 

where pa, P\, ■■■, J>n s^rs polynomials in s, of degrees to-,,, ot,, .... w,; 
respectively. The singularities of the equation are, of course, the 
roots of po = and possibly ^ = oo ; owing to the form of all 
the other coefficients, it is natural to consider* the integrals 
for lai^e values of \z\. 

It will be assumed that the integrals are not regular in the 
vicinity of z = oo . When a normal integral exists in that vicinity, 
it is of the form 

where is a uniform function of z~' that does not vanish when 
s = x , aJid O is a polynomial in z of degree (say) to, so that the 
integral can be regarded as of grade m. As in ^ 85—87, the 
value of ii' is obtained, by making the m highest powers in the 
expression 

i)oa'" + p,0'"-' + ... +p„ 

acquire vanishing coefficients; and a Puiseux diagram at once 
indicates whether a quantity fl' of such an order can be con- 
structed. The value of m — 1 is the greatest among the 
magnitudes 

provided two at least of them have that greatest value, which may 
be denoted by k. Then for such normal integrals as exist, we have 

when k is an integer, and 

where [h] is the integral part of h, when h is not an integer. The 
integrals are of grade ^h + 1, or ^ [h] + 1, in the respective cases ; 
and the equation is of rank k + 1. 

Take the simplest general case, when the equation is of rank 
unity, and when, in the vicinity of 3= co , it may possess n normal 
integrals which, accordingly, must be of grade unity. No one of 
the polynomials^, ...,pa is of degree higher than p„; assume the 
degree o?po to be k, and let 

* See Poincar^, Jnier. Joiirii. Math., t. vii (1885), pp. 30.^—268; Acta Math., 
t. vjij (188(5), pp. 295—344. 



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100.] NORMAL INTEGRALS 315 

where some (but not all) of the coofficients a may be zero and, in 
particular, where it will be assumed that a^ and «„ differ from zero. 
The determining factor for any normal integral is of the form e^' : 
6 satisfies the equation 

fr,(f) = a„^"-l-a.(9''- + ...+fl„_.^ + a„ = 0. 
The preceding theory then shews that, if the roots of this equation 
are unequal and are denoted by 8^, 9^, .... 6^, the normal integrals 
are of the form 

the quantities cr^ are given by the equations 

where 

and ^i,(jh,---, ^nare uniform functions of r"', which do not vanish 
or become infinite when s = x . Special relations among coefficients 
are necessary in order to secui'e the conveigence of the infinite 
series <p ; unless these conditions are satisfied, the foregoing 
expressions only formally satisfy the differential equation and, 
as integrals, they are illusory. 

Ei^. 1. Prove that the equation 

possesses three normal int^rals in the vicinity of j; = co , when a is a positive 
integer not divisible by 3 ; and obtain them. 

.Kk. 2. Prove that the equation 
pOBseisses three subnormal int^rals in the vicinity of x= re , when 



-(-9' 



m being an integer not divisible by 3 ; and obtain them. (Halplien.) 

.Ec, 3, Skew that the equation 

has two normal integrals in the vicinity of 3^='xi ; and, by obtaining them, 
verify that the points a;= 1, x^ ~i are only apparent singularities. 

(HaJphen.) 



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EXAMPLES [100. 

Show that the equation 

one integral, which is a polynomial in x, and two other integrals, 
normal in the vicinity of a; = to . (Halphen.) 

Ex. 5. Prove that, if normal integrals exist for the equation 

the constant a must be the product of two consecutive integers. (Halphen,) 

Sx. 6. Prove that, if all the singularities for finite values of 2 which are 
by tbe integrals of the equation 



are poles, and ii pt,Pi, ..., ^„be polynomials ins such that the degree of p^ 
is not leas than the greatest among the degrees of pi, ,.., p„, then the 
primitive of the equation can be obtained in the form 



where the constants qj, ..., q„ are determinate, and all the functions i^i, ..., 
<^ are rational fiinctions of s. (Halphen.) 

Ex. 7. Apply the preceding theorem in Ex. 6 to obtain the primitive of 
the equation 

where n is an integer ; also the primitive of the equation 

(ii) „-+'-:?'rf-('-?'+o)„.o, 

where ii is an integer prime to 3. (Halphen.) 

fii'. S. Similarly obtain the primitive of the equation 
aT5(a^-l)y'-3^-e(^+.»:i'-l)y=0, 
in the form 

(MatK Trip., Part ii, 1895.) 



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101.] Laplace's definite integral 317 

PoiNGARfi's Development of Laplace's Definite-Integral 
Solution. 

101. Several instances, both general and particular, have 
occurred in the preceding investigations in which formal solutions, 
expressed as power-series, have been obtained for linear differen- 
tial equations and have been rejected because the power-series 
diverged. These instances have occurred, either directly, in 
association with an original equation, or indirectly, in association 
with a subsidiary equation, when an attempt was made to obtain 
regular integrals of an equation, some at least of whose integrals 
were not regular; and they have arisen when an attempt has 
been made to obtain normal integrals of an equation, which is of 
the requisite form but the coefiicienta of which do not satisfy the 
latent appropriate conditions. 

In such instances, the expressions obtained for formal solutions 
do not possess functional significance. But Poincar^ has shewn 
that it is possible to assign a different kind of significance to such 
solutions in a number of cases. In particular, there is a theorem*, 
due to Laplace, according to which a solution of the given diifer- 
ential equation with rational coefficients can be obtained in the 
form of a definite integral; this solution has been associatedf by 
Poincare with the preceding results in § 100 relating to normal 
integrals. For this purpose, let 

where the contour of the integral (taken to be independent of e) 
will subsequently be settled, and T is a function of t the form of 
which is to be obtained. If this is to be a solution of our equation, 
we must have 

or, if 

r7, = 6,(»-)-6,t«-'-f... + 6„ 



* See my Treatise on Differential Equahong § 140. 

+ In the memoirs ijuoltd in the fdolnote un p 314. The following exposition 
s based paitly upon tlieie memciri, paiOv npon Picard's Cti-ars d'Jnalyse, t. ill, 



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318 Laplace's [101. 

the necessary condition is 

({U^z" + Ujz"-' + ... + Uk) e'-'l'dt = 0. 
Let 

F,-."TO.^-.'-.|(TO._,)+...+(-l)'-|a('''^.-). 
forr=l, 2, ...,h. Then 

for each of the h values of r ; and the value of [e*^ F",] depends 
upon the contour of the definite integral. Using this result, the 
above condition becomes 

[le"r,]+/.»{TO-|(m-.) + ... + (-i)>|.(m)}.i< = o, 

which will be satisfied, if T bo a solution of the equation 

and if the contour of the integral be such that 

The equation for T is 

„ d'r /, ill, „\ di-'T 



so that its singularities are the roots of (/„ = 0, that is, are the 
points d,, Os, ■■; ^n, and possibly infinity. Writing the equation 
in the form 

the value of Pj when t is infinite is - 

on. Further, the quantity S Vy involves derivatives of T up to 
order k-l inclusive. 

This equation for T has its integrals rcgnlar in the vicinity of 
each of its singularities 6^, 6^, ..., $„: their actual form will he 



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lOJ.] DEFINITE INTEGRAL 319 

considereil later. Let "^^ denote the most general integral of the 
equation for T in the vicinity of Sg; then, assuming that the 
conditions connected with the limits of the definite integral can 
be satisfied, we have an integral of the original differential 
equation in the form 



^je''-^,dt, 



and this result is true for s= 1, 2, ..., n. Now '^g is certainly 
significant, because it is a linear combination of & regular integrals 
of the equation for T; hence we have a system of w integi'als of 
the original differential equation. 

102. This system of k significant integrals can be transformed 
into the system of n normal integrals, when the latter exist. 
They can be associated with the formal expression of the n 
normal integrals, when the latter are illusory. 

A preliminary proposition, relating to the given differential 
equation, must first be established*. In the first place, let it be 
assumed that all the constants in the equation for T are real, and 
that T and t ai'e restricted to real values. That equation can be 
replaced by the system 

dt " 



When we substitute 

r,. = 0^e-*', (r=0, 1, ...,k-l), 

with the conventions that T,, — T and %„ = 0, the modified system 



^^ = - P^Q - P^-^e, - ... - P,0j._. - (P, - X) @fc. 

' It is due to Llapounoff (lfi92|i see Hoai-d, Cours d'Anabju,X. ill. p, 36: 



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320 liapounoff's [102. 

Hence 

i^(@'^ + ««),= + ... + 0v,)-?^(®=+ ©/+.-.+ ©Wj+C^-POeVi 
+ ©Hi + ©.e, + . .. + «H)s_@i_i 

- Pft 00*^, - ... - p,ei_,0fc_3 . 

Take a real quantity (o, smaller than the least real root of [/"„ = ; 
as ( ranges along the axis of real quantities between — co and t^, 
all the quantities Pj, Pa, ..., P^ remain finite. Hence, by taking 
a sufficiently large value of \, the quadratic fonn on the right- 
hand side can be made positive for that range of values of t ; and 
therefore, as t increases from — «) to i„, the quantity 

steadily increases in value. Consequently, when ( decreases from 
(j to — =o , the quantity 

steadily decreases in value. As („ is not a singularity of the 
equation, the values of 0, @i, .... 0j_i for any integral that exists 
at % are finite there ; their initial values are finite, and therefore 
each of the quantities [0], j@,|, ..., |0v_i| remains finite and 
decreases steadily, as ( decreases from 4 to — o^ . Hence the 
quantities 

all remain finite within that range, that is, no one of them can 
become infinite, for a value of X sufiiciently large* to make the 
quadratic form positive. 

Next, suppose that the constants are complex, so that T, Ti, ... 
can have complex values ; but let t still be real. Then we write 

's Differential Caieulus, Srd ed., p. 408. In the 



so that it is auiKeient to take \ greater thiin the greatest positive value which 
malies the leit-hand side in the last inequality vauisli. 



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102.] THEOREM 321 

for all values of r, where 0^ and i|r^ are real ; the system of 
equations takes the form 

t=*™. t=t.« (.=0,1 ,,-M 

Oa 0i_2 + q-i ■^i.-.i — ... — ^10 + q,,-^ 

where 0o=0, '^„ = ylr, and Ps=Ps + iqs- We now have 
equations; on substituting 

they give the modified set 

- 9^ •J' - j)j,^ 



= \ 2 (4>r' + ^r') + (X - Pi) (<['%-i + ^t"*-l) + bilinear terms. 

As before, by choosing a sufficiently large value of X, the right- 
hand side can be made always positive. Then, by taking a value 
io smaller than the least real root of ?7o = 0, and by making t 
decrease from ft, to — m , so that all the quantities p and g are 
finite, it follows that, for such a variation of (, 

steadily decreases, and therefore that each of the magnitudes 

remains finite within the range from % "^o — oo . Hence each of 
the quantities 



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322 liapounoff's [102. 

remains finite within tlie range of t from („ to — oo , for a value 
of X sufficiently large to make the quadratic form positive. 

Lastly, let the constants be complex, so that T, Ti,... can have 
complex values ; and now let t be complex in such a way that, in 
the variation from to towards — aD , where 
( = ff, -j- re"*, 

a remains unaltered. The independent variable now is t, a real 
quantity, varying from to — qo ; and the preceding argument 
applies, A finite number \ can be found such that each of the 
quantities 

remains finite within the range of (. But 



hence a finite quantity X' can be chosen, so that each of the 
quantities 

remains finite within the range of ( from („ towards - co . 
In the first and the second cases, let 

^ = X + (T, 

where rr is any real positive quantity that is not iniinitesimal ; and 
in the third case, let 

where <r is any real positive quantity that is not infinitesimal. 
Then, because 

in the respective cases tend to zero, as t becomes infinite in its 
assigned range, it follows that a quantity //. of finite modulus can 
be obtained, such that 

all become zero when t becomes infinite in its assigned range. 

This is true, a fortiori, when fi. is replaced by another quantity 
of the same argument and greater modulus. 



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102.] THEOREM 323 

lb also is true when any one (or any number) of the quantities 
T ghoiild happen to be multipUed by a polynomial- in t. For all 
that is necessary is to take a value /i+p, where p has the same 
argument as (i ; then 

e'"P, 

where P is a polynomial in t, is zero in the limit, when ( is infinite 
in its assigned range. Thus a quantity /i can be chosen so that 

where P, Pj, .... Pi_i are polynomials, all become zero when t 
becomes infinite in its assigned range from 4, which is not a 
singularity of the equation, to — co , 

103. This result is now to be applied to the equation which 
determines T. Let t=dr be any one of the roots of Ug ~ 0, and 
consider a fundamental system of integrals in that vicinity. If 

[ JJ 1 
: + (fc-l) = U77 -1. 



the indieial equation for Or is 

Suppose that p is not an integer. The integrals which belong to 
the exponents 0, 1 , .... k--2 are holoraorphic functions of t— 0^ 
in the vicinity of 0^ (Ex. 12, § 40) ; and the integer which belongs 
to p is of the form 

(t - e,y F (t - Or), 
where P is a holomorphic function of its argument. 

The contour of integration has yet to be settled. In con- 
nection with the value 0^, we draw a straight line from that point 
towards — co , either parallel to the axis of real quantities by 
preference, or not deviating far from that pai'allel, choosing the 
direction so that the line does not pass through, or infinitesimally 
near, any of the other roots of [Tj = ; and we draw a circle with 
6r as centre, of such a radius that no one of those other roots lies 
within or upon the circumference. The path of t is made to be 
(i) in the line from — co towards 0^, as far as the circumference of 



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324 DISCUSSION OF [103, 

the circle, (ii) then the complete circumference of the circle, 
described positively, (iii) then in the line from the circumference 
back towards — oo . So far as concerns the conditions imposed 
upon T by the relation 

at the limits, we have only to take the values at the two extremi- 
ties (= - oo . Now V^ is a linear ftinction oi T,T^, ..., Tt-, , the 
coefficients of these quantities in that linear function being poly- 
nomials in z and (; hence, taking z as equal to the quantity ^ of 
the preceding investigation, or as equal to any other quantity of 
the same argument as fj. and with a greater modulus, we have 

at each of the two infinities for ( ; and so the conditions at the 
limits are satisfied. 

In these circumstances, the complete primitive of the equation 
for T is 

T^A(t~e;yF{t-e,) + Qit-er), 

where Q is a holomorphic function of t — dy, involving m — 1 
arbitrary constants linearly. The corresponding integral of the 
original equation then arises in the form 

Tdt, 

taken round the chosen contour. 

104. We proceed to discuss this integral for large values 
of |s|. Let a be the radius of the circle in the contour, so that 
the series P and Q converge for values of t such that \t — 0r\<a. 
For simplicity of statement, we shall assume* that the duplicated 
rectilinear pai't of the contour passes parallel to the axis of real 
quantities from t^-dr — a to i = — x. From the nature of the 
integral T, we know that a finite positive quantity X exists, such 
that the value of 

* The alternative would be merely to take 

it value of a, Knii then jnalie t" vary from - a to -m . 



f"" 



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104.] THE DEFINITE INTEGRAL 325 

remaina finite, as t decreases from 8r — ct to — co . Let £ denote 
the maximum value within this range ; then 

for all the values of (, and then 






Let z have the same argument* as X, and have a modulus greater 
than \\\, that is, with the present hypothesis, let z be positive; 
then the part corresponding to the lower limit is zero, and we 
have 

r"'e"rdt < -^ e<'-» '"'-"• , 
J -a, z — x 

for values of z that have the same argument as X and have a 
modulus greater than |X| ; aud S is a finite quantity. 

Similarly, if, after t has described the circle, S' denote the 
maximum value of e^*2' for fl^ — a >(> — <», thou the second 
description of the linear part of the contour gives an integral, 
such that 

r "e'^Tdt < ^-^ e''-» '^-"' , 
for similar values of e ; and B' is a finite quantity. 

If, then, these two parts of the integral be denoted by /' and 
/"' respectively, we have 



where a is a positive quantity ; hence for any constant quantity q, 
however large, we have 

Limit (s^e-'^'-I') = 0, 

when s tends to an infinitely large positive value. Similarly, in 
the same circumstances, we have 

Limit (s^e-^^'F") = 0. 

* This form of atatemeiit is suited also for the variation of ( indicatecl in the 
preceding note. 



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326 poincare's discussion [104. 

Now consider the integral round the circular part of the 
contour. As Q(t — 8r) 's a holoraorphic function over the whole 
of the circle, we have 

taken round the circle ; and therefore the portion of the integral 
I e'-'Tdt contributed by this part is /", where 

/" = J (( _ e,)^ e'^P (( - 6,) dt, 

on taking A — 1. The function P is holomorphic everywhere 
within and on the circumference, so that we may take 

P(t-Or) = c, + c,it-$r) + ... + c^(t-erT + R„, 

where \R,a\ can be made as small as we please by Bufficiently 
increasing m ; for if g be the radius of convergence of P{t~ 8,), 
so that g>a, and if M denote the greatest value of \P{t~8r)\ 
within or on the circumference of a circle of radius c, where 



|ii™|<i/'- 



for values of ( such that 

it-$,\^a<c- 

The value of the integral taken round the circu inference can 
be obtained as follows. Draw an infinitesimal circle with $,. as 
centre, and make a section in the plane from the circumference of 
this circle to that of the outer circle of radius a along the linear 
direction in which t decreases towards — co . The subject of inte- 
gration is holomorphic over the area of this slit ring : and there- 
fore the integral taken round the complete boundary is zero. Let 

' T. F., % 23. 



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104.] OF LAPLACE'S INTEGBAL 327 

J' denote the value along the upper side of the slit, J" the value 
along the lower side, K the value round the small circle which is 
described negatively ; so that 

J' = r {i - OrY e"P (* - Br) dt, 

J" = /"'■"""e-s-i. (f _ OrY e'=P (( - dr) dt 

and, if the real part of p be greater than — 1, then* 

Hence, beginning at the point on the outer circumference which 
is on the lower edge of the slit, we have 

r + J' + K + J" = 0, 
that is. 

Let u denote the integral 

U= { '' it- OrY ^^i, 
and consider the value of it, for large values of z. Let 

( - f , = - T = re" ; 
then 

Taking real positive values of s, write 



so that, as z is to have very large values, the upper limit for % 
with the new variable is effectively + oo ; thus 






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328 Laplace's solution [104. 

Also, if V denote the integral 

then 

V ^ {- ly e^'- r T^ e-" R^dr 

Further, 

IJ*^ y'-e-yR.ndy \ < M (^J"^' ^/„ V^'^^y 



which, when the real part of p + 1 is positive, can be made less 
than any assigned finite quantity as m increases without limit, 
because a < c. 

Using these results, we have 

r = f' J 2 c„ (( ~ BrY + -H J (f - dry e" dt 

when TO is made as large as we please, and the real part of p is 
greater than — 1. Hence /" is a constant multiple of this 
quantity. 

105, If now Wr i^enote the integral of the original equation, 
we have 

■Wr=U'Tdt 

= /■ + /" + /'", 
so that 

For very large values of z, the first term on the right-hand side 
tends to the value nero ; so also does the second term. The third 
is a constant multiple of 

i {-\yz-'c^Vip + a + \). 



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105.] AND NORMAL INTEGRALS 323 

Hence, dropping the constant factor, we have 

Wr = e'^^s-'-' i (- l)-s-c. r (p + a + 1), 

for very large values of s. If the coefficients, of which c^ T(p+a+l) 
is the type, constitute a converging series, then this expression 
has a functional significance. If they constitute a diverging 
series, the result is illusory from the functional point of view. 
Now we have 



P + l = 



dU, 



and therefore the preceding integral, when it exists, is of the 
form 

When the aeries converges, this expression agrees with the foi'm 
in § 100, which is 

where <l>r is a holomorphic function of a"' for large valnes of 2. 

It thus appears that, when Laplace's solution of the equation, 
originally obtained as a definite integral, can be expressed ex- 
plicitly aa a function of z, which is valid for large values of ^, it 
becomes a normal integral of the equation. 

This normal integral has arisen through the consideration of 
the root S^ of the equation U^ = 0. When the corresponding con- 
ditions are satisfied for any other root of that equation, there is a 
normal integral associated with that root. Hence, when n normal 
integrals exist, they can be associated with the roots of the 
equation U^ = 0, which comprise ail the finite singularities of the 
equation in T. 

Note. It has been assumed that p is not an integer. When 
p is an integer, logarithms may enter into the expression of the 
primitive of the equation for T, and they must enter if p has any 
one of the values 0, 1, .... k — 2. There is a corresponding inves- 
tigation, which leads from the definite integral to the explicit 
expression as a normal integral. When the normal integral exists, 



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330 EXAMPLES [105. 

it can always be obtained by the process in § 100. If logarithms 
enter into the expression of e^u, they enter into the expression of 
u in the usual mode of constructing the regular integrals of the 
equation satisfied by !(. 

Es:. 1. The preceding method of obtaining the normal integral gives a 
test as to the convergence of the series in its espr^sion. If the infinite 



converges, which must be the case if the expression for the developed definite 
integral is not to prove illusory, its radius of convergence r is given by the 
relation* 

Lim|c„r(p + a + l)r = ^. 

But, from Stirling's theorem for the approximation to the value of r In), 
when n is infinitely large, we have 



%+c^{t-6;,+c,,{t-e;f-\-... 

must conven^e over the whole of the i-plane ; and therefore the integral TV is 
of the form 

wtere ^ (() is holomorphic over the whole plane ; a I'esult due to Poincard 

Ex. 2. Prove that, if the condition in Ex. 1 ia satisfied, a normal int^ral 
certainly exists. (Poineard.) 

Ex. 3. Consider Bessel's equation 

for large values of |i;|. The int^rala in the vicmity of j; = oo may be 
normal — they are not regular — and, if normal, must bo of gradi" unity. 
Accordingly, let 

then the equation for a is 
We take 



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105.] bessel's equation 3 

and theu seek for a regular integral (if any) of the equation 

j^m" + {^ + 20^) u' + (to -v?)u = 0. 
If an integral, regular in the vicinity of a: = =o, can exist, it is of the form 



Substituting, and making the coefficients in the resulting equation vanish, 
we have 

c^{26p + 6) = 0, 
anii, for all values of m, 

«=j + So„, + i(2p-2m-l)-0. 
and the latter then gives 

Hence, taking 60 = 1, and 6 = i, a formal solutioa of the original equation is 

and taking S= — i, C|,=l, another formal solution is 



If 2ji is an odd integer, positive or negative, both series terminate ; and the 
forma] solutions constitute two normal integrals of the equation. It is not 
difficult to obtain an espression given by Lomaiel* for ■/„, in a form that 
is the equivalent of 

IfSn. is not an odd int^er, both aeries divei^o ; and the formal solutions are 
then illusory as functional solutions. 

When Laplace's method of solution is adopted, so as to give an integral of 
the form 

the equation for T is 



On writing 



(i^ + l) T" + ZtT' + {l -^ n^) T=0. 
V independent variable, the equation for T is 



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BB8SELS EQUATION AND 



[105. 



This is the differential equatioi 
elements are given by 



e function, wliose (Gau 



o + fi + l=3 



a^=l 



Thp contoui cf the integral oonsi^tR of (i) a <,irde ruund i as tPiitre with 
radius leas tlnti 2 ('o as to exclude — ;, the otlier finite singuHnty of the 
equation m T), and then (ii) a duphcited hue fium a point in the Liruum 
feience pissing m the direction of a diameter tontinued towards co The 
ai^ument of / and the argument of j: muit he -^uth that the real pajt of xt 
is negative. In order to construct the integral, we need thf complete primi- 
tive of the ^-equation in the vicinity of ii = 0: it is 

where A and B are arbitrary constants. The part multiplying A, being a 
holomorphic function, merely contributes a zero term to w ; and we need 
therefore substitute only the other part. Manifestly, we may write B=l, 
Now 

^(a-y + 1, j3-y+l, 2-y, ^) = i^(<i-i, (3-^, h ") 



1 2m + l 



, „ 2m + 






n(m+^)-n(-^); 



Taking this value of c^,, we substitute 

in the definite integral. In the preceding notation, we have 
d.=i, p=-i, <r,= -(p + l)=-i 
n(m-<r,) = n(m + 4); 
so that, when the solution 



where t=i-2iv, is expanded into explicit form, ifc becomes a constant 
multiple of 



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105.] DEFINITE INTEGRALS 333 

But 

^n(,H)^ "«'-"'»'-:l-'"-*''-'' n(-i). 

SO that, after substituting for c« and rejecting the constant factor n(-^), 
the integral becomes a constant multiple of 

m=o ni! {2iJ^)"" 

which agrees formally with the expression earlier obtained. 

The corresponding integral, associated with the primitive of the ^-equation 
in the vicinity of i= — i as a singularity, can be similarly deduced*. 

E^: 4. Shew that the equation 

where «,'^ - ia^ is not zero, can be transformed to 

xy/' + {\, + \ + 2)'y/ + {x+i (\i^\ti)}w = 0. 

Assuming Xi, X^i ^i + ^a "^ot to be integers, prove that the latter equation is 
satisfied by 

for an appropriate contour independent of x ; and deduce the normal series 
which formally satisfy the equation. (Horn.) 



Double-loop Integrals. 

106. Before proceeding further with the investigation in 
§§ 101 — 105, which is concerned partly with the precise determ- 
ination of a definite integral satisfying the linear differential 
equation, we shall interrupt the argument, in order to mention 
another application of deiinite integrals to the solution of certain 
classes of linear equations. It is due to Jordanf and to Poch- 
hammerj, who appear to have devised it independently of one 

' In connection with the aolation of Bessel's equsition by mes^ne of definite 
integtals, papers by Hunkel, Math. Ann., t. i (1869], pp. i67 — 501; Weber, ib., 
t. 3X3V1I (1890), pp. 404—416; Macdonald, Proc. Lond. Math. Soc, t. sm (1898), 
pp. 110—116, ib., t. XXI (1899), pp. 165—179; and the treatise by Graf u. Gubler, 
Einleitung in die Theorie der Bessehchen FunUionen, (Bern), t. i (1898), t. n (1900) ; 
may be consnlted. 

t CoiiT-B d'Aiwlyu, 2= k&., t. Ill (1896), pp. 240—276; it had appeared in the 
earlier edition of this work. 

+ Math. Ann., t. xsxv (1890), pp. 470—494, 495- 536; II., t. sxxvii (18S0), 
pp. 500—511. 



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334 nouBLE-Loop [106. 

another. A brief sketch is all that will be given here : fur details 
and for applications, reference may be made to the sources just 
quoted, and to a memoir by Hobson*, who gives an extensive 
application of the method to harmonic analysis. 

As indicated by Jordan, the method is most directly useful in 
connection with an equation of the form 

where Q (s) and zR (s) are polynomials, one of degree n, the other 
of degree ^niiiz, R {z) also being a polynomial. For simplicity, 
we shall assume Q (s) to be of degree n. 

Consider an integral 

W= { T {t - zY+''-' dt, 

where 2" is a function of t alone ; this function of ( has to be 
determined, as well as the path of integration. We have 

AW-(-l)"(a + 7i-l)(a + n- 2). ..(« + ]) 
+ {t-zY\R{z) + {t-z)R'{z)^^^''R"{z)+JATdt 



= /[« (* - ^y-' Q (i) +(t-zrii im Tdt, 



the summation being possible because Q and R are polynomials 
of the specified degrees. The integral will be capable of simplifi- 
cation, if the integrand is a perfect differential ; accordingly, 
we choose T so that 



which gives 



TE(t)=^[TQ{t)\, 

Q(t) 

* Phil. Tmne., 1896 (A), pp. 443—531. 



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106.] INTEGRALS 335 

The preceding integral then becomes 

jdV, 
where 

mt) 

Hence the original differential equation will be satisfied if 
jdV^O; 

and this will be the case, if the path of integration is either 

(i) a closed contour such that the initial and the final values 
of V are the same : or 

(ii) a line, not a closed contour, such that V vanishes at each 
extremity*. 

Each such distinct path of integration gives an integral. It ia 
proved by Jordan that there is a path of the first kind, for each 
root of Q ; and that, when there is a multiple root of Q, paths of 
the second kind are to be used. 

Again, restricting Q (s) for the sake of simplicity, we assume 
that each of its n zeros is simple; let them be (ti, esg, ..., a^- As 
the polynomial R (z) is of degree less than n, we have 



M(fl 



= S - 



where 7,, ..., 7,1 are constants; and then 

To obtain the paths desired, take any initial point in the plane ; 
from it, draw loops^f round the points a,, ..., On, z, and denote 
these by A^, A^, ..., An, Z. Take any determination of 

* A third possibility would arise, if the patli were 
same value at its extremities; tut thia case is of yery n 
■t T. F.. % 90. 



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DOUBLE-LOOP INTEGRALS [106. 



which is the subject of integration in W, as an initial value ; and 
let the values of W, for the various loops J.,, ..., j4„, Z with this 
as the initial value, be denoted by W {a^, ..., W{a^, W{z) 
respectively. 

An integral of the original differentia] equation will be ob- 
tained, if the path of integration gives to F a final value the 
same as its initial value. Such a path can be made up of 
ArAgAr~'^Ag-^, that is, first the loop Af, then the loop Ait, then 
the loop Ar reversed, then the loop jig reversed. Let W{ar, Wj) 
denote the value of the integral for this path ; then W{aT, a^ is 
a solution of the differentia! equation. Taking the above initial 
vahie (say /(,) for /, we have 

W{ar, a,) = W(ar) + e"^->r Wia,) - e="^*. W(ar) - W(a,) 
= {1~ e^'v,} W(ar) - {1 - e-^r] W (a,) ; 

for after the description of A^, the initial value of 7 is e^'^t./^ 
for the description of Ag-, it is e''"'>'r+''s'7t, for the description of 
Ar~', and it is e""^*./,, for the description of j1,~\ 

It is clear that 

[l-e'^yi] W (a„ ar) ^ [1 - e"^-'.} W (a„ at) + {1 ~ e'"^,] W{at,ar); 

and therefore al! these values of the integrals, for the various 
appropriate paths, can be expressed linearly in terms of any n of 
the quantities W(«r, Cs)> in particular, in terms of 

W(z,a,). W(z,a,), ..., W(z,a^). 

Each such quantity is an integral of the original equation ; and 
we therefore have n integrals of that equation. 

iVofe. For the special cases when a or any of the conetantB y is an 
int^er ; for the cases when § (I) has multiple roots ; and for the oases when 
B(t) ia of degree n — \, while Q{1) is of degree less than m — 1 ; reference may 
be made to the authorities previously cited. As already stated, all that is 
given here is merely a brief indication of the method of double-loop integrals. 



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106.] EXAMPLES 337 

Es:. 1. Consider the equation of the qu.irter-pei'iod in elliptic functions, 

Here we have 

9W«(--l), 






-/'- 



t''{t-rf=r-'{t-\)'^ 



■(i-iy^(t-z)--'dl, 



and tho path of integration has to be settled. 
We have 



W(lj==2J'dW, 



I dW, 
where a mai'ks the initial point of the loops. Hence 

H'(0, l) = 2W{0)-2W{l) = ij' dW, 
W[0, 0-2tf(0)-2F{2) = 4prfir; 
and thu.s two integrals of the equation are given by 

l^dW, tdW. 
The comparison with the known results is immediate. 
Ea;. 2. Integrate in tho same way the equation 
,, „, rf% ^ dw , 
' ' cb= dz 

whore a and 6 are constants. (This is another form of the equation 

(l-,.)2'-2(»+l).f+(.-«)(»+« + l),.-0, 

by Hobson {I.e.) for unrestricted values of the constants m and re.) 
V. 22 



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338 ASYMPTOTIC! [106. 

Ex. ;i. Prove that when the equation 

wheie /( i» I tuuittnt, 15 iiiiijectpd to thf tr wistorm xtion 

the tunstonned equation (nhich is of Fuchsnn ty]ie, § 54) can, under a 
LCitain condition, be trcat<yl bj the foregoing method ind issuming the 
condition to be ^atiified ohtaiii the mtegial 

Ej 4 ^pply the method to the equation 

apply it alao to the equation of the hypergeometric series. 

(Jordan ; Pochhammor.) 
Ex. 5, Apply the method to solve the equation 
{\-i')ie"-2zw'-lw = 0, 
for real values of s such that — 1 <3<1. Sliew that the equation is ti-aus- 
formed into itself by tiie relations 

(^-l)(ir-I) = 4, w{z + \f^W{Z+lf- 
and deduce the solution for real values of z such that 1 < z < gd . 

(Math. Trip., Part ir, 1900.) 



PoiNCABit's Asymptotic Representations of an Integral. 

107. After this digression, we resume the consideration of 
the investigations in §§ 101—105. 

In those ca.ies wheu the infinite series in a normal integral 
diverges, the normal integral has been rejected as illusory from 
the functional point of view. There are, however, cases belonging 
to a general class which, vfhile certainly illusory as functions of 
the variable, are still of considerable use in another aspect : they 
are asymptotic representations of the integral, to use Poincar^'s 
phrase*. 

A diverging series of the form 



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107.] KEPRESENTATIONS 339 

is said to represent a function J(x) asymptotically when, if 8n 
denote the snm of tho first ra + 1 terms, the quantity 
„•{!(,) -S.\ 

tends towards zero when x increases indefinitely ; so that, when a; 
is sufficiently large, we have fl;"{i/"(a:) — S„) < e, where |e| is a 
small quantity. The error, committed in taking S„ as the value 
of J, is less than 



Lch smaller than 



that is, the error in taking Sn as the value is much smaller than 
in taking S„_i. (The definition, though stated only for large 
values of x, applies also to the vicinity of any point in the finite 
part of the plane, tnatatis mutandis.) 

The asymptotic representation is, however, not effective 
for all values of the argument of the independent vaiiable. If 
a;" {-/(«) - Sn) tended uniformly to zero for all infinitely large 
values of x. the function J(x) would be holomorphic, and the 
series would converge: the permissible values of the argument 
of the independent variable are therefore restricted. It is manifest 
from the nature of the ease that, when such a series is an 
asymptotic representation of a function, the series can be used 
for the numerical calculation of the approximate value of the 
function for large values of a: with a permissible argument: the 
error at any stage is much less than the magnitude of the term 
last included. Without entering upon any discussion of the 
question why a diverging series, which is functionally invalid, 
can yet, when it is an asymptotic representation of a function, 
be of utility for the numerical calculation of the function, it is 
proper to mention one conspicuous example of the use of such 
series, as found in their application to dynamical astronomy*. 

The normal series, derived from the solution of the equation 
as represented accurately by the definite integrals, are proved by 
Poincar^ to give this type of asymptotic representation of the 

* In particular, see Poinoare, HUcaniqiie Cileste, t. ii. 



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340 NORMAL INTEGRALS AND [107. 

solution. For, denoting the solution by w, and tlie sum of the 
first m -t- 1 terms of the series 

by Siji, we have 

Now 

c 
where 

and 

iHl<l. 

Then, as before, we have 

r M u 

7 0™+' J _li-Pr| 
c 
which is a multiple of 

by a quantity independent of a. When we take 

so that, as a is to have large values, the limits of y effectively ai'e 
to + CO , the last definite integral is a multiple of 



-I T — •T^y'-*''^"^e-ydy. 



This definite integral is finite. Denoting its value by 7, we have 

where a is a quantity independent of e, and 7 is finite. Hence, 
when 2 is sufficiently large, we have 

z™ (we~=*'-s''+' - Sm) < e, 



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107.] ASYMPTOTIC EXPANSIONS 341 

where | e | is a small quiintity ; and so we can say that S^ asym- 
ptotically represents we~'^'- £'•+'■, or we can say that the normal series 
is an asymptotic representation of the actual integral, the repre- 
sentation being valid (on the hypotheses adopted earlier) foi' large 
positive real values of ^. 

Note. For further diseuasioii of these asymptotic expansions 
in connection with linear differential equations, reference may be 
made to Poincar^'a memoir*, which initiated the idea. Among 
other memoirs, in which the subject is developed and new applica- 
tion.9 are maile, special mention should be made of those^f by 
Kneser, and thosell by Horn. Picard's chapterj on the subject 
may also be consulted with advantage: and a corresponding dis- 
cussion on integration by defiuite integrals is given by Jordan§. 

Ex, 1. Shew that the complete primitive of the differential equation 

in the vicinity of ;c = co , can be asymptotically represented by 
('^ + "' + ,^^+-)«>«'^+(^o+t + 5 + -)^'"'*'' 

and hq, (9o ive arbitrary constants. (Kneaer.) 

Ex. 2. In the differential equation 

i(-*S)+(«+<')»-°- 

ifc^ is an arbitrary parameter, A, B, C are real functions of x and (with their 
derivatives) are holomorphic when ai^x^h; moreover, A and B are positive. 
Prove that a,ii int^ral of the equation, determined by initial values that are 
independent of h, is a holomorphic traoseeiidental function of k ; and shew 
that, for large values of k, its asymptotic expansion if of the form 

,!,.(*,+*>+.^.)co.i,+(*l + J' + .,.).mi», 

whore ^a, (f",, <^j, .,., mi are functions of x, (Horn.) 

* Acta Math., t. vni (1886|, pp. 295—344, 

+ Crelle, t. oxvi (1896), pp. 178—213; ib., t. txvii (1897), pp 72— lOS; lb., 
t CM (1899), pp. 267—275 ; Math. Ann., t. xlix (1897), pp. 383—399. 

II Math. Ann., t. xm (1897), pp. 432—473, 473—496 ; i6., t. l (1898). pp. 525— 
556-, ib., t. LI (1899), pp. 346—368; ib., t. LH (1899), pp. 371—392, 340-362. 

X Coura d'Analyse, i. iii, ch. xiv. 

% Goiirs d'Analyse, t. iii, ch, n, § iv. 



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342 RANK [107. 

Ex. 3. Shew thiit the equation 

has a solution of tlie form. 

where A^, Bj„ ace rational functions of k, and that it has an aaymptotic 
solution of the form 

(*.+|= + ...)o™fa+(^ + *J + ...).mfa, 

and indicate the relation of the Molutions to uue anofclier, (Poinearri : Horn.) 



Equations of Rank greater than Unity replaced by 
Equations of Rank Unity. 

108. When the differential equation 

possesses, in the vicinity of 3 = O) , normal integrals which are of 
grade m, then, denoting the degree of the polynomial p^ by ■m^, it 
follows (aa in § 85) that the degree -m,. of the polynomial p, is such 
that 



the sign of equality holding for some at least of the < 
Also, if e" be the determining factor of any such integral, then 
Of is the aggregate of the first m terms in the expansion, in 
descending powers of z, of a root of the equation 

The existence of the normal integral then depends upon the 
possesion of regular integrals by the linear equation in u, where 

In the case where in = 1, the method of Laplace certainly gives 
the integrals of the differential equation, even wheu the normal 
series diverge ; but it is not applicable, when m is greater than 
unity. Poincare, however, devised a method by which the given 
equation is associated with an equation of grade unity: Laplace's 
method is applicable to the new equation, so that its primitive is 



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108.] OF EQUATIONS 343 

known : and from tliis primitive, an integral of the original 
equation can be obtained by means of one quadrature. The new 
equation is of order ra'"; and the investigation leads to an 
expression for 

w dz ' 
which, whefl it oxiata, can be obtained more directly by Cayley's 
process (§ 92), 

Poinear^'s method is as follows. Let the given equation be 
supposed to possess n normal integrals of grade m, say, in the 
form 

e"'<''.^(4 B"'^'>M^) e""'«0«(^); 

let these be denoted by /i (a), /^(j), ...,/„ (a). 

Let a denote a primitive mth root of unity, say e™ ; and 
consider, in connection with any integral /(z) of the original 
equation, a product 

!,="nV(«'^). 

Then y satisfies an equation of order ii™. which possesses ?^™ 
normal integrals 

/.(')/. («)/.(»'^) .■•/.(«--H 

where a, b, c, ..., k are the numbers 1, 2, ..., n or some of them, 
any number of repetitions being permitted ; and these normal 
integrals are of grade m. Lot 

and let the equation for y be 

where, if Q^ be of degree in z, then the degree of Qjc_r in 
general is equal to d + r{m—'l), because of the grade of the 
normal integrals. Owing to the source of the quantity y, which 
clearly is not changed if 2 be replaced by sa*, s being any integer, 
it follows that the equation for y must remain substantially 
unchanged, when this change of variable is made ; hence 
fc--r(s«')g-'-'^-'"" _ 

where \ is independent of r. 



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344 BANK CH. .>'GED [108. 

Now let the variable be changed from z to x, where 

then, because 

for all values ef «, the coefficients c„. being numerical, the equation 
for 2/ takes the form 

■where 

The degree of -fi^y-j in ^, as it is determined by the highest terms 
in Q,v-a. i*5 

which is independent of 3; so that the degree* of all the 
coefficients R is the same. Further, wc have 

B,_, (.«■) - i^ c„„ («')'->'-.'-'>--'e„_(««') 

for the power of a is 

thus 

Hence the equation is substantially unaltered, when z is replaced 
by 20* in the coefficients Jt ; hence, multiplying by a power of s, 
say z", where 

« 4 f + if (m - 1) = (mod m), 

i£ becomes a uniform function of x, when we substitute 



" Some mij-ht haye vanishing ooelflcieiits in particular cases; the argument 
deals with the genei'al ease. 



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108,] HY TRANSFORMATION 345 

The new equation is therefore an equation in the independent 
variable x such that all its coefficients are uniform,- They ail are 
of the same degree, so that it is of rank unity ; it has normal 
integrals, and some of its integrals may he subnormal. Laplace's 
method can be applied to this equation ; and we then have a 
solution in the form of a definite integral. 

The way in which this definite integral is used, in order to 
bring us nearer a solution of the original equation, is as follows. 
Let 



'.=/(^a"). 



(s = 0, 1, . 



-1), 



This has to be differentiated N(=n™) times, derivatives of w„, 
Wj, ..., w,„_i of order n being replaced, whenever they occur, by 
their values in terms of derivatives of lower order, as given by 
the diffei'ential equations which they satisfy; and, from the iV"+ 1 

equations involving y, -j^ , ,.., -j-^, the iV" products 



de^ '"rfs* ■ 



d^ 



where a, b, ..., k « 



1 can have the values 0, 1, . 



,. , ., ..., k each can have the values 0, 

eliminated. The result is the equation for y. 

involving y, ^, ..., -^E^ can be regarded 



products of the type 



The N equations 
i giving these N 






"^~d^ 



each in I 
such be 



i of derivatives of y and the variables. Let two 



Assuming p known, as an integral of its own equation, the value 
of lUo is derivable by a quadrature. If y, first obtained as a 



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346 POINCAR^'S METHOD [108. 

definite integral, can be evaluated into a functionally valid normal 
integral, it is of the form 

The function $ is linear in y and the derivatives of y, so that, 
when we substitute the value of y, we have 

ivhere "I* is^ free from exponentials ; and then 
1 dw^ _-^ 

which can be expressed as a series in terms of z. The exponent 
to which it belongs is easily seen to be an integer, owing to the 
form of ■!> ; thus 

1 <^M;n ™ , ^, „ a,„ «„,+i 

j-=(l(,3"'-' + ((l2™-^+ ... +(tm-i+ — H ->+-■■■ 

Wa dz Z Z^ 

But if y cannot be evaluated into a functionally valid normal 
integral, there may be insuperable difficulty in dealing with the 

quantity — . 

In instances, where the actual expression of a normal integral 
{if it exists) is desired, the process is manifestly cumbrous: as 
it does not lead to explicit tests for the existence of normal 
integrals, the simpler plan is to adopt the process indicated in 
|§ 85 — 88, which gives either a normal integral or an asymptotic 
expression for an integral in the form of a normal series. 

For further consideration of Poincare's method, reference may 
be made to his memoir, already quoted, and to a memoir by 
Horn*, who discusses in some detail the case, when the linear 
equation is of the second order and of rank p. 

Ex. 1. In the case of an equation of the second oi-der which is of rank 2, 
saj 

shew that, if w = 0(«), and if w^i^^{^-x), which will satisfy the equation 
d^w, , , , dw, 

i-«-^><-'>7Er+''><-"''"'-»' 

- Asia Math., t. xxiii (1900), pp. 171—201. 



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lOS,] EXAMPLES 

then ;i vai'iablo y., whore 



uniquely in terms <iiy. 

Tf, liowever, the invariants of the two equations are equal, so that 






of a quadratic equation, the coefficients of which are expre^ible in terras 
of y. (Horn.) 

TJx. 2, Discuss the equation 

for lai^e values of a-. (Poincare.) 

Ex. 3. Shew that, in the vicinity of r— o^ , the equation 

ormal integral of the second grade, when a is an odd positive 



Es^. 4. Ohtain the normal integrals of the equatioi^s 
(i) :k^" = (^ + |)2/, 

(ii) ^y = 2j;(l + &^)y + (;^^6V-26a;-j:)y, 
in the vicinity of a^=o; , 



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CHAPTER VIII. 

Infinite Determinants, and their Application to the 
Solution of Linear Equations. 

109. In the investigations of the present chapter, infinite 
determinants occur. These are not discussed, as a I'ule. in books 
on determinants ; a brief exposition of their properties will there- 
fore be given here, but only to the extent required for the 
purposes of this chapter. Their first occurrence in connection 
with linear differentia! equations is in a memoir* by G. W. Hill : 
the convergence of Hill's determinant was first established f by 
Poincare. Later, von Koch shewedj that the characteristic method 
in Hill's work is applicable to linear differential equations generally; 
with this aim, he expounded the principal properties of infinite 
detenninants§. The following account is based upon von Koch's 
memoirs just quoted, and upon a memoir]| by Caazaniga, 

Let a douhly-infioite aggregate of quantities be denoted by 



where i, k acquire all integer values between — co and + x ; the 
quantities may be real or complex, and they may be uniform 
functions of a real or a complex variable. They are set in an 

* First published ill 1977; republished 4c (a Math., t. viii (1886), pp. 1—36, 
t BvU. de la Sw. Math, de France, t. xiv (1886). pp. 77—90. 

I Acta Stath.. t. sv (1891), pp. 53—63 ; ib., t. nvl (1892—3), pp. 217—235. 

§ Foi tocther cliscuBBion of theii properties and their applications to linear 
differential eiiuations see a memoii by tbe same writer, Aeta Mat},., t. xstv (1901), 
pp 89—122 

II Anwih di Uai mat a *5er 2' t jcxvi (1897), pp. 143-218. Other memoirs 
by CazzaniKa lealin^ with the la.iai' subject, are to be found in that journal, 
&ei ' t i|18JS| pp s-i-14 S^ > t. II (1899), pp. 329-238. 



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109.] INFINITE DETERMINANTS 349 

array, so that all the quantities with their first suffix the same 
occur in a line, the values of k increasing from left to right, and 
all the quantities with their second suffix the same occur in a 
column, the values of i increasing from top to hottoni. We then 
have an infinite determinant, which may be represented in the 
form 

Constrnct the determinant An.n. where 

then if, as m and n increase indefinitely and without limit, i),n,» 
tends to a unique definite value D, we regard the infinite determ- 
inant as converging to the value D. In all other cases, the 
infinite determinant diverges. To secure this convergence to a 
unique definite value D, it is sufficient that, when any arbitrary 
small quantity S has been assigned, positive integers M and N can 
be found, such that 

!-0«.+p,n+s-0™,«|<s, 

for all values of m greater than M. for all values of n greater than 
iV^, and for all positive integers p and q. 

The aggregate of all the quantities for which i = k, that is, of the 
quantities ..,, a_i,_i, (X„,o, Ch.i, as they occur in their place in the 
determinant, is called the principal diagonal, sometimes briefly 
the diagonal ; and a constituent of reference in the diagonal, 
naturally chosen in the first instance to be aj.u. is called the 
origiv. 

Let 

then the infinite determinant converges, if the doubly- infinite 



converges, all values of i and k between — <» and + oo occurring ii 
the summation. To prove this, let 



p„..= n 1+ s |4,.,i 



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350 COUVEllGENCE OF [109. 

and consider 

Let P,„,B be expanded ; by omitting suitable terms and chauging 
the signs of others, we obtain Dm,». Hence, taking I'm,.!, making 
al! the terms positive, and adding certain other positive terms, 
we obtain .?„,„. Similarly^ we can pass from D^^p^n+q to 
Pm+p,n+q- Now take i>m+p,n+9 — -Om,!! ; make all the terms 
positive, and add certain other positive terms, and we have 

|i).+,.„^,-A.,«|<|-P™..,«-H,--^..,«l- 

But, because of the convergence of the series 



the product P^.n converges when m and n increase without limit ; 
hence, assuming any arbitrary positive quantity S, however small, 
integers M and iV" can be determined such that 

Pm-tp,n+q - P'm,n < ^i 

for all values of to greater than M, for all values of n greater than 
N, and for all positive integers p and q. Consequently, for the 
same integers, we have 

and therefore the infinite determinant converges. 

Such a determinant is said* to be of the normal form. AU 
the determinants with which we have to deal are of this type. 

Next, the origin may be changed in the diagonal without 
affecting the value of the determinant. All the conditions for 
the convergence of the determinant with the new origin are 
satisfied; let its value be D', and let D be the value with the 
old origin. Then taking any small positive quantity 5, we can 
determine integers M and N such that 

|-t'--Om,«|<S, |i>'-i>V,„,|<^, 

' von Koch, Acta Math., t. xvr, p, 221. 



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109.] INFINITE DETERMINANTS 351 

for all values of m greater than M and all values of n greater than 
N, the determinant D'^^n, being the same as il^^n, ao that, if agg 
be the new origin, m, = m- $, Wi = n + ft Manifestly, D^^^ can 
be chosen so as to include the new origin. Hence 

\D-n'\ = jD - »™,„- (D' - Z>V,,„)| 
<\D-D,„„\ + 'D'-})\„.,\ 



so that, in the limit when 8 is made infinitesimal, 

d = b: 

Similarly, the value of the determinant changos its sign when 
two lines are interchanged, and also when two columns are inter- 
changed: so that, if two lines be the same, or if two columns be 
the same, the determinant vanishes. Further, if the determinant 
be changed, so that the lines (in their proper order) become 
columns and the columns (in their proper order) become lines, 
the principal diagonal being unchanged, the value of the determ- 
inant remains unaltered. If, in any line in a determinant of 
normal form, each of the constituents be multiplied by any 
quantity ft, the value of the determinant is multiplied by /j. ; 
likewise for any column, and for any number of lines and 
columns, provided that the product of all the factors (when 
unlimited in number) converges. 

Further, if all the constituents in any line of a converging 
normal determinant be replaced by a set of quantities of modulus 
not greater than any assigned finite quantity, the new determinant 
converges. In the determinant D, let the line ao^j (the constitu- 
ents occurring for values of k) be changed, so that a^^j is replaced 
by *'t, where 

kll < A, 

A being finite ; and let D', !>'„_ a for the new determinant 
correspond to D, D„^„. For comparison with Z*'„,„ construct 
a product P,„,,(, where 

p,.,,..'n'|i + l|^,.,i|, 

1 having all values from —n to +m, except i = 0. Then, when 
i>',„_,i is expanded, there occurs in P^.n Jt term corresponding to 



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352 PKOPBKTrES OF CONVERGING [109. 

every term in D'm^n. tbe latter having gome one factor x^ that 
does not occur in ^m.n\ hence 

I term in i?;„,„|^^ jterm in P™,„|. 
Now some of the terms in D'™,„ are negative, while all the terms 
in Pm,-n. are positive; and terms arise in Pm,n. the terms corre- 
sponding to which do not occur in D'm,n- Hence 

Similarly, 

where h can he chosen as small as we please, becanse 

i |i + I \AiA 

is a converging proiluct. 

The resnlt, which is due to Poincare, is thus established. 



Properties of Oonvergisg Infinite Determinants. 

110. The development of an infinite determinant can be 
deduced from the preceding properties. We have 



-n I "tji, -11+1 • ■■■! 'hn.t 
= 2 ± (»_„__.„«._„ ^.,,_„+l . . . am.ni) 

say. In this expanded form, let 

ai_i = l + Aij, ai^k'^-^w (' + *); 
and let every term in the new expression be changed, so as to 
have a positive sign and so that each factor is replaced by its 
modulus. The resulting expression is greater than |Sm,„|; and 
every term that occurs in it is contained in P,„,«, where 

_ m ( m ) 



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110.] INFINITE DETEEMINANTS 353 

Also, Pm,« contains other terms, all of which are positive ; thus 

|S„,,|<-P,..,. 

Similarly, _ _ 

for all positive integers p and q. But P,„,„, with indefinite 
increase of m and m, is a converging product; hence 2m,n. in 
the same limiting circumstances, converges absolutely. Thus the 
usual method of development of a finite determinant holds in the 
case of an infinite converging determinant of the normal form, and 

«-[»»] l]-Z} 

= 2...a_2,j,,tt_,,p,ao,y,aj,g,as,g^ ... 

(_lY..+ (Pi-2) + (J'i-" + (J>o-"> + Wi-il + i5)-"l+- 

the sura being extended over all the permutations 

• ■■. p.., Pi, p«, ?i. 9^, ■■■ 
of the integers 

.... -2, -1, 0, 1, 2, .,.. 
Writing 

for all values of i and k, we at once have the expansion 

Z) = l + 2Ji,( + 2|^(,i, Aij\ + t\Ai_i, Ai,j, ^i,i ] + ..., 

^Aj^i, Aj^jl -4j,i , J-jj, A}^ie\ 

the summations being for all integer values from — gc to + co such 
that 

i<j<k<.... 

111. It follows from the preceding expansion of a converging 
determinant D of normal form that, when a constituent o^j enters 
into any term of the expanded form, no other constituent from 
the line i or from the column k enters into that term. Taking 
the aggregate of terms {each with its proper sign) into which Oi^ic 
enters, theii- sum may be denoted by ra; ^a^j ; and the determinant 
may be represented in the form 



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354 MINORS [TU. 

or in the form 



The quantity Wj^i is called the minor of an, and sometimes it i 
denoted by 



It can be derived from D by suppressing the line i and the 
column k, or, what is the equivalent in value, by replacing Oj^j 
by 1, and every other constituent in the line i or in the column k 
or in both by 0, and then multiplying by (— 1)'"*. Manifestly, we 
have 

It is an immediate corollary that 

0= S aj^tai.k, I 

(»4=i)[: 

k=-'B I 

for the right-hand side in the iirst is equivalent to i> with the 
line * replaced by the line j, so that the latter is duplicated ; and 
in the second, the right-hand side is equivalent to D with the 
linej' replaced by the line i, so that the latter is duplicated. 

More generally, if, in the lines 

and in the columns 

ft. A ft-, 

we replace all the terms by 0, except «a„B,, «o„p,, --., aa,.,e,., each 
of which we replace by 1, and then multiply by 

the result is the coefficient of 



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111.] OF FINITE ORDKB 355 

in jD. It nianifcstly is a minor of order r; and it is denoted by 



Clearly all the minors of any finite order are determinants of 
normal form, converging absolutely. 

If D is not zero, some at least of tlie minors of constituents in 
any line must be different from zero, and some of the minors of 
constituents in any column also must be different from aero. 
Similar results, when D ia not zero, hold for the minors of any 
order r of finite determinants, which are constructed out of r 
selected lines and any r columns, oi' out of r selected columns 
and any r lines. 



Further, the minor 



r + l. ..., 0, 1, 
r+1 0, 1, 



tends to the value unity, as r and s increase. To prove this, let 

Q,,~n{l + X\A„]]. 
where the product is for all the values of p, and the summation 
is for all the values of q, that are excluded from the ranges 
p = — r to + s, q — ~rto-^s. 

Expanding the minor, and changing every term so that its sign is 
positive and each fsictor in the term is replaced by its modulus, 
we have a new expression every term of which is contained in the 
expanded form of Qa,>-', and Qg^,. contains other terms. Further, 
the expanded minor contains the term +1 as does Q,^,., and all 
other terms involve the quantities A ; hence 

|(::; :::5::;;; 3 -i|< «..-'. 

But the product 

uh + iiA^A 

converges ; and therefore, when any small positive quantity S is 
1, integers — r and s can be determined such that 

Qs.r - 1 < S. 



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356 EXl'AJiSION Of [111. 

Taking these as the integers defining the minor, we have 
80 that 

i-s<|(:::::;:;::::;:):<i+s- 

Moreover, as integers s', r are chosen, greater than s and r and 
gradually increasing, the quantity 

decreases ; and thns the minor tends to the value unity as r and s 
increase. 

One or two properties of minors may be noted. We have 

\k, l) \k, l) \l, k) \l,k)' 

for the changes from one of these expressions to another are 
equivalent to an interchange of two lines or an interchange of 
two columns, each of which changes the sign of the determinant. 
Similarly for minors of any order. 

Again, expanding ai^^ by reference to constituents of a column, 
we have 

and expanding it by reference to constituents of a line, we have 
Similarly, 



!, because it is 
me; also 



when q is neither k nor I, because it is a minor of the first order 
with two columns the same ; also 



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111.] INFINITE DETERMINANTS 



whon h is neither * nor j, because it is a i 
with two Hnes the same ; and 



357 
.■ of the first order 



\k, I) 



= 0. 



where h is neither i nor j, and q is neither k nor I, because it is a 
minor of the first order with two columns the same and two hnes 
the same. Similarly for minors of higher order. 

The similarity in properties between finite determinants and 
converging infinite determinants of normal form is not exhausted 
by the preceding set : in particular, infinite determinants can be 
multiplied, and determinants framed from minors of an infinite 
determinant are connected with their complementary in the 
original, exactly as for finite determinants. The simpler of these 
properties are contained in the following examples. 

Ex. I. If 
are oonvei^ing doterminants of normal type, itnd if 



for all values of i and t, then 



c-[»...l 



s a convoi'ging determinant of normal type, and 
AB = C. 
Ex. 2. If ai,i, denote the minor of o^t in the determinant 

%k.' ■■■' %k. 
"i-k,' *■■' %i; I 

with the preceding notation for miDora of order ;■. 
Ex. 3. In connection with the determinant 



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S5S INFENITE DETERMINANTS AS [111. 

prove that 

Qfe|)+©(t»)-^©(*;S-(U;l)''^ 
QO;:S+©(U)*(')(l;i)-C:*;:t)"' 

and, more generally, that 

L f%\fH J H 1 .... ^ \_(hii h, h, ■■■! ir\ A 
.iAU\h-.i, K*ii, -, -f-n-i/ W. *i- ^, ■-. V 
where, in the typical term, k^, it„ + i, f« + 2, ..., ifcn-i preserve the aame cyclical 
order as fc^, ifcj, ij, ..., i^- 

In the first of these, the right-hand side vanishes if i is equal to ^, or l;^ i 
in the second, it vanishes if i is equal to t, or 4 ; in the third, it vanishes if 
k„ is equal to any one of the quantities tj, ^3, ..., i,.; and so in other cases. 

112. The infinite determinants which arise in the discussion 
of linear differential equations have, as their constituents, functions 
of a parameter p. The preceding results are still valid, if the 
condition that 

is an absolutely converging series is satislned; in particular, the 
determinant converges absolutely, and its value may be denoted 
by D (p). The parameter may be made to vary ; and then it is 
important that the convergence of D{p) should be not merely 
absolute, but also uniform, in order that it may be differentiated. 
Suppose that, in any region in the p-plane, all the functions 
Aij(p) are regulai' functions of p, such that the series 

converges uniformly and absolutely. For all values of p within 
that region, any small quantity B can be assigned, and then 
integers M and N exist, such that for all integers m^M, and 
integers —n^ — N, 

11 '1 A.cj{p)\<&. 

By analysis that follows the earlier analysis practically step by 
step, wo then infer that, for all integers m'^M, n'^N, and for all 
positive integers p and q, and for all values of p within the region 
indicated, we have 

-D,„,,.+,<p)--D„,.(p)|<28; 



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112.] FUNCTIONS OF A PARAMETER 359 

SO that l>(p) converges uniformly. Hence, within the domain 
considered, I) (p) is a regular analytic function of p. 

The expansions of D (p) in terms of its constituents have been 
proved to converge absolutely, by comparison with the expansions 
of Pm,w. where 



converges uniformly and absolutely, Pm.n is a product that con- 
verges uniformly with indefinite increase of m and n. The 
corresponding modifications in the investigation lead to the 
conclusion, that the expanded I'orra of TJ(p) converges uniformly 
as well as absolutely. 

Moreover*, this expanded form can be differentiated, and its 
derivatives are the derivatives of D (p). In particular, we have 

dp 9ffl,-,i dp 

= Z2, a. . -^^ . 
■ dp 

Thus if D vanish for a value p' of p, and if all the first minors of 

D vanish for that value, we have 

^1 = 0, 

while -;r^ is not iniinite; the first derivative of the uniform 

dp 
function D vanishes, and therefore p' is at least a double zero of 
B. In that case, we have 

d^D -^ d^ai I- .p.-^- 9a; 1 3"; t dai , 



j-i-So... 



dp dp 



-s^ss(';{ 



9af,t dojj 



Hence, if all the second minors of D vanish for that valui 
we have 

dp- 

' The proof is aimilur to those given for preceding propositions ; see to. 



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360 INFINITE SYSTEMS [112. 

and so p' is at least a triple zero of D. And generally, if all 
minors of all orders up to r — 1 inclusive vanish, but not all 
minors of order r, when p — p, then p is a root of D in multiplicity 
r ; and B is then said to be of characteristic r. The quantity r 
cannot increase indefinitely, for we have seen that minors of 
sufficiently high order tend to the value unity, so that the general 
vanishing of all minors of the same order is possible only for finite 
orders. 

But it need hardly be pointed out that the converses of these 
results are not necessarily true: thus p = p' might be a double 
root of D, while not all the first minors of D would vanish. 

113. The purpose, for which infinite determinants are to be 
used in this place, is in connection with the solution of an un- 
limited number of equations, linear in an unlimited number of 
constants. Let 

and suppose that the infinite determinant B, where 

converges uniformly ; it is required to find the ratios of the 
quantities x to one auother which satisfy the equations 

Ui = 0; (i = - ^ to + 00 ), 

the quantities a; being themselves finite, so that we have 

where X is finite. 
We know that 



fG)- 



converges absolutely; its value is B when j = k, and is when / 
is different from k. Moreover, the series S<\s is an absolutely 

converging series, and hence for values of o! considered, we have 



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113.] OF EQUATIONS 361 

where TJ is finite. Hence, by one of the propositions already 
establishei^, the quantity S, defined by the equation 



also converges absolutely, so that 

for all the other terms give a zero coefficient for x. Hence, if 
Mi = for all values of i, and if we are to have values of ic^ different 
from zero, then 

D = Q, 

which is a necessary condition. We shall assume this condition 
to be satisfied. 

If some at least of the first minors are different from zero, 
then the equation 

shews that any one of the quantities u, which it contains, is then 
linearly expressible in terms of the others, and so the correspond- 
ing equation w = is not an independent equation. Let m, then 
be omitted on this ground ; we have 

?(i,'iV=ffG; ")"'■'''■'■ 

where on each aide the summation is for all values of i except 
1 = 0. The coeificient of x^ on the right-hand side is 

?(i,' ")"•■'■ 

This is zero, if q is different from both k and I; it is 



= I ; and it i; 



=fu: 



Jc. V 



?((,'*)«"-"•■'■ 



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362 

if g = k. Thus 



INFINITE SYSTEMS 



[113. 



U i) 



= -«ll,t^i + «(,(■»* ■ 



But all the quantities v^ vanish ; hence 

We thus have 

for all values of /; and ^ is any finite quantity, for only the 
ratios of the quantities x are determiuabe. 

Similarly, if D be of characteristic r, so that the minors of 
lowest order which do not all vanish are of order r, let 



he snch a minor different from zero. We then have 



Thus the coefficient oi x 



When q is equal to any one of the integers ^-,, ^i, ..., 0r, this 
coefficient is equal to a minor of order r — 1 and so vanishes. 
When q is not equal to any one of those integers, the coefficient 
is equal to a determinant with two columns the same, and it is 
therefore evanescent. Hence 

,S = 0, 
and therefore 



:::;:)" 



-T 



A. .... /3„, A, /3„ 



■:::.> 



where, on the right-hand side, m must not be equal to any one of 
the integers a„ ...,ar. It thus appears that there are r relations 
among the quantities m; and that, in particular, each of the 
quantities m.^, u,^, ..., ■«„,. is linearly expressible in terms of the 



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113.] OK EQUATIONS 363 

remamiiig cjiiantitiea u. Accordingly, we assume these r quanti- 
ties u omitted from consideration. 

Denoting by a any integer other than a, a,, and by any 

integer other than ^j, ..., ^r, we have 



=(*:*::.■:;«:)'"'-&",&;:;;:»:) '''-(A.sT.r.'.A)'''-' 



in the same way as for the simpler ease ; hence, as all the quanti- 
ties !t„ vanish, we have 



U.A J"'-.!, (a. A A._ 



SO that all the quantities xg a,rc linearly expressible in terms of r 
such quantities. 

For further properties of infinite determinants, reference may 
be made to the memoirs quoted at the beginning of | 109. 

Ai'j'LicATioN TO Differential Equations. 
114. When the differentiai equation is given in the form 



the substitution 

~l /'^•''^ 
W = we ■' 

leads to an equation of order n in w, which is devoid of the term 

involving -r— ^ • The coefficients of the new equation are linearly 

expressible in terms of Q^, Q,, ..., Qn-i, Qn, and the expressions 
involve derivatives of Qj up to order n — 1 inclusive and integral 
powers of Q,. We may therefore take the differential equation in 
the form 



^w-i.^ 



' (/s"-= "■ " ' de 



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364 INFINITE DETERMINANTS APPLIED TO [1.14. 

We assunie that, in the vicinity of 3 = 0, it possesses no synectic 
integral, no regular integral, no normal integral, and no subnormal 
integral. The point ^ = is then a singularity of the coefficients ; 
and, if it be only an accidental singularity {of order higher than s 
for Pg, in the case of some value or values of s), the conditions for 
the existence of a normal integral or a subnormal integral are 
not satisfied. We assume the coefficients P still to be uniform 
functions of s, and we shall suppose that their singularities are 
isolated points. Let an annulus, given by 

Ji<\z\< R, 

be such that its area is free from singularities, no assumption 
being made as to the behaviour of the coefficients P within the 
circle of radius R; then it is known* that each such coefficient 
can be expanded in a Laurent series 

P.= ic,,^^^ (r = 2, 3, ..., n), 

which converges uniformly and unconditionally within the annulus. 
Without loss of generality, it may be assumed that 

Ji<l<li': 

for, otherwise, we should take a new variable Z = e(RR')^, and 
the limiting radii R and R' of the annulus for Z then satisfy 
the conditions 

R<l<R'. 

Further, owing to the character of the convergence of P^, we 
have 

dP, - 

d ( dPr\ ? 

and so on; all these series converge uniformly and unconditionally 
within the annulus. Hence also 



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J 14.] LINEAR DIFFERENTIAL EQUATIONS 365 

similarly converges within the annulus, where R (fi) is any poly- 
nomial in /J. ; and therefore, taking the circle I e\ = ]., every point 
of which lies within the annulus, the series 

_i|iiWc,,„j 

converges, 

115. From the general investigations in Chapter ii, it follows 
that the equation certainly possesses an integral of the form 

J/=ZP<P (3), 

where p is any one of the values of g— . logo), the qnantity w 

being a root of the fundamental equation associated with an 
irreducible (but otherwise simple) closed circuit in the annulus ; 
and the quantity is a uniform function of z. As the integral is 
not regular, the number of negative powers of 3 in is unlimited ; 
and so we may write 

In order to have an adequate expression of the integral, the 
quantity p must be obtained ; the value of a„ h- a,,, for m = + 1, 
± 2, ..., + X , must be constructed; and the resulting series must 
converge for values of s within the annulus. 

We first consider the formal construction of the expression for 
the integral. Let 

<i>(p) = p(p-l)...(p-n-t-l) + c,,_,p{p-l)...(p-n + S) 

+ Cs,-3p(p-l)...(p-n + 4) + ... + C„^,,-n+iP + Cn,-,,; 

+(p + /i)...(p+>(-m + 4)c,,r_^_5+ ... 

...+(p+/J.) C„-.,r-«-„+, + Cn.r-^-n ; 

and write 

Gm(p)=0(p + m)a™+rC^,^a^, 

where, in the last summation, the values of fi are from — x to 
+ CO, with /j. = m excepted. Then we have 

p(,) = i G,(rt ..*--, 



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INTEGRAL TN THE FOEM OF 



[115. 



SO that y is an integral of the differential equation if 

for all values of m from — oo to + <» , there being no assumption 
that the negative infinity is the same numerically as the positive 
infinity. Let 

n 



for all values of fi other thai 
■^in.m with the convention 



») 



ti; and introduce a quantity 



ffm (p) = (m + p) S -^ni. 



where the summation now is for all values of y:*. We then requi 
the infinite determinant 



"((>)-[*.,,] 



■ - ■ T ''i''— 1 — y ) ^ » ^—1 * ^— I 1 ' '^—1 ' 

..., ^1,-2 1 ''/'"l,— 3 ■ ''/'"!, II I 1 ' '"/''l,! 



the necessary and sufficient condition of the convergence of which 
is the convergence of the double series 

for all values of m and fi between — co and + oo except m = f*. 

116. In order to establish the convergence, we firat transform 
the expression of Cm.,.- Let 



then we may take 

(p+t^)(p+^l-l)...(p+^l-p + '\) 

= (p + m-X){p+m-\-l)...ip + m- 
= (p + my + «p,. (P + "O"- + V. ip + '«)' 



\-p + \) 



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116.J A LAURENT 

where a,,,, is a polynomial in X of degree r. Using this for all the 
terms in G^ „, we have 



where 

A,(X) 




-+A,{\)(p 


+ «.).- + ... 


+ i,.W, 




Accordingly, w 


e have 










^2S 


<^ (p + m) 1 


W,(x)j + SS 


!*((> + »•) 1 


!^.W! + 




Now the series 




'i" |fl{i)o,. 


+ S2t : 


byk" 


(Ml- 


converges for 


every value 


2, S, ..„ n 


of r, where 


! ii(\) is 


any 



polynomial in X. Hence 

every term of which (for the various values of p) converges, because 
««-ii,s-j!+s is a polynomial in X of degree s — p + 2, and therefore 
the whole of the right-hand side is a converging series. Accord- 
ingly, we may write 

'F A,{X) = H„ (s = 2, ...,n), 

and then each of the quantities |ffs| is finite. 
We thus have 

'.(p + mTjl ,|(p + m)^ 



i!?t..,.|<i^.i!rii^h™!ri^i 



+ ... 



"' . I * (p + m) 

Assuming p to be any quantity, different from any of the roots of 
any of the equations 

*(p + m)-0, 



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368 INFINITE DETERMINANT [116. 

each of which is of degree n, we know that all the series 

converge absolutely, for the values k—2,S, .... n. Moreover, the 
sum of each such series is a function of p : and then, if p varies in 
a region no point of which is at an infinitesimal distance from any 
of the roots of ^(p + m), the convergence of the series is uniform. 
Accordingly, the double series 

converges uniformly and unconditionally ; and therefore the infinite 
determinant ii(p) converges uniformly and unconditionally, pro- 
vided p does not approach infinitesimally neai- any root of any of 
the equations 0(p + m) = O. Clearly, il(p) is a uniform function 
of p, for such values of p. 
Further, we ha.ve 

^«.,.(p)0(p+m) = C,„,„(p), 
and therefore 

t^+,,^, ip)<f>(p + m + l) = t?^+,.«+, (p) 

= -.|r,„,^(p + l)^(p + l+m), 
so that 

Construct the infinite determinant 0.(p + I), and then replace 
each constituent -^m.^-ip + 1) by ^,„+,^„+,(p); the result ia to give 
the modification of il (p), which arises by moving eaoh column one 
place to the right and by depressing each row one place, in other 
words, by taking ^}r,^,(p) in the diagonal as the origin instead of 
ifpo „(/)). But such a change makes no difference in a determinant 
which converges absolutely ; we therefore have 

n(p + i) = n(p), 

or the infinite determinant fl is a periodic function of p. 

Lastly, by making p infinitely large in such a manner, that it 
does not approach infinitesimally near any of the roots of any of 
the equations 

.^(p + m) = 0. 



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116.] MODIFIED 369 

(which roots for different values of m differ only by real integers, 
80 that if we take p=u + iv, where u and v are real, it will be 
sufficient to take v large), we reduce to zero every constituent 
that lies off the diagonal of il(p). As every constituent in the 
diagonal is unity, and every constituent off the diagonal is zero, it 
follows (from the law of expansion of an absolutely converging 
determinant) that 

Lim n (p) = 1, 

provided p tends to its infinite value in the manner indicated. 



Modification of the Infinite Determinant ii (p). 

117. It is convenient also to consider another infinite determ- 
inant associated with fl (p). The equation G^ ip) = was taken 
in the form 

<l>(m + p)Xir^.^a^ = 0: 

and the infinite determinant ii(p) was composed of the constituents 
^m.f If ^"^ infinite determinant were composed of constituents 
Ip {m + p) yJTm,^, then the row determined by the integer would 
have a common factor 0(m4-p); and thus there would be an 
infinitude of factors, the product of which either should converge 
or should be made to converge. Let pi, p^, .,., p„ be the roots of 
(j) (p) = 0, so that 

<f> (p) = (p - pi) (p - p„) . , . (p - p„), 
and therefore 



^. 



To change this into a form suitable for an infinite convergin 
product, we multiply by 



with the convention 

S.(rt-1. 

As A^(p) remains finite and is not zero for finite values of p, v 
may replace the equation (?m(p) = by 



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370 MODIFICATION OF THE [IIT. 

Now let 

for all valuPB of ?« except m = 0, and 

Xo.'^ n (p-p„); 
also let 

Then the equations between the constants a have the form 



In association with these equatio 
determinant 



consider the infinite 



(p)= 


X- 


J 














X-^-3 


X->-i 


x-». 


X-. 


. X-l!. •■■ 






X-1.-2 


X-J.-1 


X'V 


X-. 


. X-.,.. ••■ 






X.,-. 


X«,-i 


X",o 


X... 


, x... , ... 






»,-. 


X>.-i 


»,. 


Xi.i 


. Xi.= . ■■■ 






Xt- 


%2,-l 


X.,. 


X=,i 


. Xs,B . ■■■ 



Taking the diagonal to be ..., X-^.-s, X-i.~" X'>.'" X'." %',!' ■■■• ™ 
require to establish, (i), the convergence of the series 

summed for all values of m and /*, except m = fi, from - ;» to + k 
and (ii), the convergence of the series 

S(x-,.-l), 

in order to know that the infinite detenninant D (p) converges. 
We consider first the double series %'Zxm,u- ^s* 



i,(p) = .n-*„(f).n. 



(m + 0). 



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117.] INFINITE DETEHMINANT 371 

The quantities p^, p-i, ■■-, pn are finite; hence, so long as p remains 
within a finite region that does not lie at infinity, there is & finite 
quantity K which is larger than any value of j/'m(p)| for values of 
p within that region. Hence, as 



.\<K'- 



when m is not zero. When m is zero, we have 

»,,=*.WC.,,-o..,. 

Proceeding exactly as with tlie series SS^m,^. in § 116, summing 
for all values of m other than zero, and for all values of fi other 
than m = /i, we have 



lf + " 



-+|fflX 



lp + » 



■\tl.\t- 



every term of which is finite, and therefore 

is finite. Also 

l\(!..A^\HJ\p'-\ + \H.\\p'-'\ + 
which is finite, so that 

converges. Hence the double series 

Slimmed for all values of m and /t between — co and + c 
, converges. 



Moreover, all the series, which 
superior limits in the inequalities, converge uniformly withir 
region of p considered ; hence the double series converges 
formly. 

The establishment of the convergence of the series 

i(fc,.-i) 



except 
in the 



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372 CONVEEGENCE OF [117. 

is simple. We know, by Weierstrass's theorem*, that tho series 

converges uniformly and unconditionally; so that, if 

»„,,. = x.,--i. 

the infinite product 

n(i + (?™,™) 

converges uniformly and unconditionally; and therefore^ 

n_(i + |9.,,|) 

converges. But 

n(l + |^™,,„|)<l+2|^,„,,„|; 

hence S |^m,inl converges uniformly, that is, the series 

_S(x,.,«-l) 

converges uniformly and unconditionally. 

The eoiivergonce can also be established as follows. Let 



=(-^'). 



and choose a finite positive integer p, such that, for values of p under c 
sideration, we have 

\p-p'\<P, 
where p' is any one of quantities pi, p-2, ..., fn- The sum of the terms 

J(x«.-I) 

is finite, and may be omitted without affecting the convergence: and we e 
sider the sum of the remaining tenns, for which we have 

\m]>p. 
We have 



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117.] 




THE DETERMINANT 




and therefore 












2>ii^|7'^|<| 
<- 


|.j / 1 p - 


f-' + -- 


-Pff^ 




\p-i'.r\^ 




»? " ^ 




1 \9-pA- 














Now for all the values of m under consideration 




and therefore 


J |p-P"[ 


-^1 ip-P'L, 


- p+l p+l' 






^.K^P^ 


-P,\\ 




so that we maj 


'take 


'■-S-"- 


-?^)\ 




where 




1f.|<i. 






Hence 


Xm. 


„=nM^ 







ttj 2,,{,-,,l" 



Now 2 (iir(p—p^)2 ia finite for all the values of puuder conaidenttion, and 

it is finite for all values of m if ji^ involves m; let ^denote the gieateat value 
of its modulus. Again, for any quantity 0, we have 



othat 



r writing 



A' jV! 1 iV=i ] 



<-i4'(4)"4:(4.)^ 



shewing that the si 
converges. 



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374 EVALUATION OF THE [117. 

The infinite determinant I)(p) thus converges uniformly and 
unconditionally for all values of p in the finite part of its plane. 
Its relation to li (p), which converges similarly for values of p 
that are not infinitesimally near any of the roots of any of the 
equations ^{p + m) = 0, is at once derivable from its mode of con- 
struction from n(p). The row of quantities Xnt.j'lp) ^^ ^(p) f*^^ 
the same value of m is derived from the row of quantities i^m.^ in 
il (p) for that value of m, through multiplication of the latter by 

h^(p)(f>(m + p). 
Hence 

D(p)-a(p)nj,.(p)^(„,+p) 

where 

ii(p)=nK(p)<l-{^ + p) 

-Jt [i{(' * 'if"-') " " '""']] .i <" " "•'• 

and 11' implies multiplication for all values of m between + oo and 
— cc except m = 0. Also 



n(/')=^'^j;Msm(p-p.)'rl. 

Now D{p) has been proved to be finite (that is, to be not 
infinite) for all finite values of p ; and manifestly, from its form, it 
is a uniform funct.ion of p, so that it is a holomorphic function of 
p everywhere in the finite part of the plane. Further, D,{p) is a 
uniform function ofp; and it has been proved to be not infinite 
for values of p, which are not infinitesimally near any one of the 
roots of any of the equations (p(p + m) = 0, the aggregate of all 
these roots being 

pi + m, p^-i-m, .... pn + 'm, (i« = — oo to + oo ). 

Hence, owing to the relation 

D{p)-n{p)n(,p). 



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117.] DETERMINANT 375 

it follows that these roots are poles of ^(p)- Take a line in the 
p-plane inclined at a finite angle to the axis of real quantities, 
choosing the inclination so that it does not pass thi'ough any of 
the points p^ + m for all values of <r and m ; let it cut the axis of 
real quantities in a point f. Take the point /4-1 on that axis, 
and through it draw a line parallel to the former, thus selecting 
an infinite strip ia the /j-plane. Since 

a{p + i)^a(p), 

the uniform function il{p) undergoes all its variations in that 
strip: and within the strip, we have 
Lim fi (p) = 1. 

Owing to the nature of the poles of il (p), the strip contains n of 
them, which may be regarded as the irreducible poles : suppose 
that they are pi, p^, ...,/3«. Within the strip, p= oo is an ordinary 
point of the simpiy-periodic function ii (p) ; it follows* that the 
number of its irreducible zeros is also n, account of possible 
niuitiplicity being taken ; let these be /j/, />/, ..., p«'. Hence 

n(p) = A ^^^ t(P - Pi') '^j si n {(P - PaO ' ^1 - ■ ■ sin {( p - Pn) ■^] 
sin i(p - pi ) wj sin Kp - p, ) tt) ... sin Kp - p„ ) tt) ' 

taking account of the holomorphic character of -D(p) for finite 
values of p, and of the relation 

Here, A is independent of p. To determine A, we use the 
property 

Liran(p) = l, 

which holds for 

p = u+ iv, 

in the Hmit when v is infinite, whether positive or negative. 
Taking v positive and infinite, we have 

and taking v negative and infinite, we have 



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376 FORMATION OF [117. 

Hence XpJ — Sp„ is an integer ; if it is not zero, we can make it 
zero by substituting, for the quantities p',, values congruoiit with 
them. Assuming this done, we have 

2;>/ = Sp„, 

^ = 1, 
so that 

ilip)'- 

and therefore 



Moreover, the quantities pt, p-i, ■■■, pn a^e the roots ol ^{p)~ 0, ao 
that 

hence 

Sp.'-i»(«-l). 

118. Next, we consider the expression 



»in((p- 


-Pi. 


|T| 


|...sin((p- 


-Pn 


)■'] 


sinUp- 


-Pi" 


)ir\ 


|...sin(((>- 


■Pn 


M' 


DM- 


: n 


.h. 


iKpj-i'W 







=IQ-^ 



i to prove that this series converges for all values of z 
within the annulus. It manifestly arises from D (p), on replacing 
;\;o fc in D (p) by 3* ; we shall therefore assume that F is transformed 
into this modified shape of i)(p). When the determinant is in 
this shape, we multiply the column associated with m by z~™, and 
the row associated with m by z™] these operations, combined, do 
not change %m,ra, and they do not alter the value of the determ- 
inant. Let this combined pair of operations be carried out for 
all the values of m from —x to + co ; the result is to give a 
determinant, which is equal to Y and has 

Xv,,"~' 
for its constituent in the same place that ;^j,^, occupies in 1) {p). 
Hence, as for D (p), so T converges uniformly and uncondition- 
ally for values of p within the p-region selected, and uniformly 
and unconditionally for values of z within the annuius, if the 
doubly- in finite series 



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118.] AN INTEOBAL 377 

converges uniformly and unconditionally within those regions, 
and if 

S(x«-1) 
converges uniformly and unconditionally. 

The latter condition is known to be satisfied, owing fco the 
convergence of J) (p). It remains therefore to consider the con- 
vergence of the double series. 

With the notation of §§ 115 — 117, we have 

Now 

A^ (\) s' = a;„_^,. C;, x_5 3^ + a„_3, ,_, c,, x_, j^ + ...+ Cr. w sh- 
owing to the definition of the coefficients in the original differen- 
tial equation, the series 

converges uniformly and unconditionally, for values of a within the 
annul us 

R<\2\<B'; 
and therefore the series 

converges uniibrmly and unconditionally for the same range. 
Denoting this by J^, we have 

and \Jf\ is not infinite for any of the values of z. 

Again, as (§ 117) 
and 



when m is not zero, we have 



■*.W%r^^ 



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378 CONSTRUCTION OF [118. 

Proceeding with the double series SSxni.M-^'""^ exactly as with 
the double series £x"i.f > omitting for the preaent the terms corre- 
sponding to m = 0, and remembering that the summation is for all 
values of m other than m =/i, we have 



<K^J 



every group of terms in which is finite, so that 

SS|x,,,^»-| 

is finita Also, taking account of the terms omitted for the value 
m — 0, we have 

S |c.,,^'| < \J,: If'-", + i./.| |p"-; + ... + IJ-J, 

which is finite, so that 



summed for all values of m and ^ between — oc and + co except 
m = /i, converges unconditionally. Moreover, all the series which 
occur in the supenor hmits in the inequalities converge uniformly, 
both for the values of s considered and the retained range of p ; 
hence the double series converges uniformly and unconditionally. 
The proposition is therefore established for 

£©"■ 

A similar investigation shews that the series 

for any value of r, the numbers a and /S being any whatever, 
converges uniformly and unconditionally for values of z within 
the annulus, and for values of p in the range that has been 
retained. 



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119.] 



CONSTBUCTION OF IrREGULAE INTEGRALS. 



119. These results may now be used, by a generalisation of 
the method of Frobenius in Chapter iii, to construct expressions 
for the integrals of the equation 



Writing 

1/= 2 (I™ 3''+'^, 

and adopting the notation of § 115, we have 

-e,(p) «»+'-», 

it GmW-O, 

for all values of m between — oo and + oo , except m—i. The last 

equations are equivalent to 

/i„<p)e»(p)-o, 

that is, to 

for all the values 0, + 1, + 2, ... of m, except m, = i. Let 

We have 
that is, 

for all the values of h. Hence, writing 
o...Af'V 



we have 
and 






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380 IRREGULAR [119. 

Thus the quantity y, where 

^-^-!©^'" 

satisfies the equation 

The determinant D (p) is of normal form ; the series for y con- 
verges uniformly and unconditionally, alike for vahica of s within 
the annulus R<\z\<R', and for values of p within the finite 
region contemplated. 

120. Let p=p' be an irreducible simple root of D(/j) = 0. 
Then the first minors of constituents in any line cannot vanish 
simultaneously for p — p'', for 

and the left-hand side does not vanish for p = p'. Selecting minors 
of constituents in the line i, we havo 



and 

that is, ^1 13 an integral of the equation. 

Similarly for any other irreducible simple root of D {p) — 0. 

121. Next, let p = p' be an in-educible multiple root of 
i> (p) = of multiphcity a. 

Firstly, suppose that some of the first minors of D(p) do not 
vanish for p = p'', let some of these non-vanishing minors be 
minors of constituents in the line i. Then, in the vicinity of 
p = p, we have 

as a quantity satisfying the equation 

P (2/) = ^sM-i-" (p - p-f R(p- p'% 

where Rip — p") does not vanish when p — p. It therefore follows 
that 



^^-ip-p'y-'^^i^'P'P')' 



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121.] INTEGRALS 

SO that, if ^< 0" — 1, we have 



\dp''),~/ 



is an integral of the equation. Hence, corresponding to the 
irreducible root p' of multiplicity a, there are integrals 

^'^^\h 0] ^'^'' "^ ^" ^"^ ^ = ''^ + ?/« i«g ^^ 

3/. = 2 r|^, Ql zf-^" + 2^, log ^ + y, (log s)= 
= 9jj + 2iji log s + y, (log ^)^ 

y.-.^7),_i + (^-l)7,,„,log^ + ^- ''~y^^^ -"^^^.^(log^)' + ... 

... +(<r- l)j?,(log3)^-' + yoaog^)'"'. 
when, in each of these expressions on the right-hand side, we take 
p = p. 

122. Next, still taking p = p' to be an irreducible root of 
i> (p) = of multiplicity a, suppose that, of the minors of successive 
orders, those of order r are the first set which do not all vanish 
for p — p'. Let the lowest multiplicity of p' for first minors be o-,, 
for second minora be o-^, and so on up to minors of order r — X, the 
lowest multiplicity for which is denoted by a-,-,. Then, owing to 
the composition of B in relation to first minors, to the composition 
of first minors in relation to second minors, and so on, we have 

<r><ri>(r,>...>o-^,. 
There are two ways of proceeding, according as r < ff, or r = a-. 

First, let r < cr. With the preceding notation, wo have 

and 



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382 SUB-GROUPS OF [122. 

After the explanations given in the construction of these expres- 
sions, we know that p= p is a root of m.ultiplicity o-i for some 
of the minors in the expression for y. As before, in § 121, the 
quantities 

dy 3^ 

^' dp' ■■•■ dp--'' 

when in each of these we take p — p', are such that 

for A, = 0, 1, ..., <r — 1. But owing to the fact that p = p' ys, a root 
of all the minni-s I ,1 of multiplicity ctj, all the quantities 
dy d-^y 

■'' dp' ■■■■ dp-'-' 

vanish when p — p'- Hence the n on -evanescent integrals which 
survive are 

dp"' ' 3p''i+^ ' ""' 3,3'^' ' 
when p = p'. They have the form 



J,,. = aI^ Q «'« + (a, + 2) ,„ log . 
and so on : their number being 



Next, p — p' is, a root of least multiplicity o-i for some of the 
minors of the constituents of any line i: and there mast be at 
least two such minors. For 



i-w-sQ-v 



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122.] IRREGULAR INTEQBALR §83 

if p = p' 13 & root of multiplicity o-j + 1 for all the minors but 
[ , ] , then, as it is of multiplicity ^ .Ti + 1 for Z* (p), it would be of 

multiplicity o"! + 1 for f J. Similarly for any other line. Once 
more substituting 

in P(y), we have 

= t?i (p) z'-+'-" + Gj (p) z<-+^-^, 
provided G^ {p) — 0, 

for all integer values of p from -co to +x except p — i, p=j. 
The last equations are equivalent to 

kp(p}G^(p)=0, 
that is, to 

for all integer values of p except t and j. 
Consider quantities ag of the form 

for ail values of 0, tho quantities A and B being arbitrary. With 
these expressions for a^, we have 

Each of the sums on the right-hand sides vanishes, when p is not 
equal to either i or j : and thus the preceding expressions satisfy 
the equations 

«,((>) e,(p)-o, 

for all integer values of p except i and j. Further, 

*,(p) ft w ='?»,,«, 






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384 SUB-GROUPS OF [122. 

and, similarly, 

i,(rte,(rt=-^(;)+i)Q. 

Using these values, we have 
as the expression for y ; and it satisfies the relation 

i* iy) = Gi (p) ^"^"" + Gj (p) 3^+j-» 

As the right-hand side of the last equation has p^p' as a root of 
multiplicity a-j, the quantities hi(p) and hj(p} having no zero for 
finite values of p, it follows that 

-(PI.-' 

for X = 0, 1, ..., ffj — 1. Therefore all the quantities 
dy d"'-'y 



when p = p', satisfy the equation P (w) - 0. Owing to the form of 
y above obtained, which has p = p' rr a root of multiplicity <rj, all 
the quantities 

^' dp' ■■■' dp"'-' 
vanish when p — p. Therefore the surviving in1 



'"-if- 

,d so on : their number being 



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122.] IRREGULAR INTEGRALS 385 

Similarly for the next sub-group. With the same notation as 
before, we have 

P (y) - G, ((>) ^'+'-" + B,W «'+'■- + a. (p) ^■+»-", 
provided 

for all values of^, other than i,j, h, from — « to « . The analogy 
of the preceding case suggests 

for all values of 6, where A, B, C are any quantities. With these 
expressions for oe, we have 



=AX 



Each of the three sums on the right-hand side vanishes, when p is 
not equal to either i or j or k: so that the preceding expressions 
for a# satisfy the equation 

e,(/>)-o, 

for all values of p other than i or j or h. Further, 

Thus 

where ^(e, p) is a linear combination of minors of the second 
order; and 

the coefficients a^ being linear combinations of 
third order. 



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386 SUB-GROUPS [122. 

As *(2, p) has p=p' as a root of multiplicity a^, it follows 
that 

for X = 0, 1, ..., C72 - 1 ; HO that all the quantities 

^' dp- ■■■' ap"-' 

when p — p', satisfy the equation P (w) = 0. Owing to the form of 
the coefficients a^ in y, each of which has p^p' as a root of 
multiplicity a-^, all the quantities 

dy d"_^y 

^' dp' ■■■' dp"-'' 
vanish when p~p' \ and we therefore are left with the integi-als 



and so on : their luiniher being 



Proceeding in this manner, we obtain successive sub-groups of 
integrals ; the total number in the whole group is 



which ia the multiplicity oi p = p' as & root of D (p) = 0. 

123. Two cases, both limiting, call for special mention. 

It is manifest that, if tr — o-i > 1, the first sub-group contains 
integrals whose expressions involve logarithms; likewise for the 
second sub-group, if ffi — (ra>l; and so on. If, then, all the 
integrals belonging to the multiple root p = p' of Z* (p) = are to 
be free from logarithms, we must have 

0-- 0-1 = 1, o-j-o-,-1, .... 



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123.] OF INTEGRALS 387 

and there fort! 

which thus is a limiting case of the preceding investigation. 

An intimation was given that, when r = o", a ditiferent method 
of proceeding is possible. As a matter of fact, the property of the 
infinite system of lineHi relations, ostablished in § 113, leads at 
once to the result. Let 



be one of the non-vanishing minors of order r belonging to D (p) ; 
then 

and the quantities a.^^, a^^, ,.., a^^ are bound by no relations, so 
that they are arbitrary constants. The integral determined by 
these coefficients is 

it manifestly is a linear combination, with arbitrary coefficients 
«■>.,, •■■- C'Mr, oi r integrals which are, in fact, the group of integrals 
above indicated. 

The other limiting o;\se occurs when r = 1 : all the <r integrals 
belong to a single sub-group. In that case, there exists at least 
one minor of the first order which does not vanish when p=p'; 
the condition is both necessary and sufficient. 

124. We thus have a set of rr integrals, belonging to an 
irreducible root p' of D(p) = which is of multiplicity o-. 
Similarly for any other irreducible root of D{p) = 0; hence, when 
all the irreducible roots are taken, we have a system of n integrals. 
We proceed to prove that this system of integrals is fundamental. 

For, in the first place, it follows (from the lemma in § 27) that 
the integrals in any sub-group are linearly independent, on 
account of the powers of log^ which they contain. 

Next, there can be no relation of the form 
Ci2/.. 1 + O^y^ , + ... + C^yr.i = 0, 



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388 FUNDAMENTAL SYSTEM [124. 

with non- vanishing coefficients 0. If such an one could exist, the 
coefficient of every power of z in the aggregate expression on the 
left-hand side must vanish. Writing 

i, j, h, ... =Pi, Pi, Pi, ■■-, Pr\ 

k, I, m, ... =5,, q„ q„ ..., q^ ' 
we have 



"-M'e. 



'PuP2,--;pA , (Pl,p2, 



'P„P^, 



and the quantities Ag,, A^^„ ..., j4s.s f're at our disposal. Let 
tiiese last be chosen so that 

Then the coefficient of sf'+'i in ^s,i is zero if (<s, and it is different 
from zero if i = s: let it be denoted by [ye,,\- 

The above relation being supposed to hold, select the co- 
efficients of 3^'+^', 2*'+?', .... z*''+9r in turn. As they vanish, we have 

0. b.,i + c. [*.]., + ... + c, [».,]„ = 0, 

from the coeSicient of sf'+^^ ; every terra vanishes except the first, 
and [yi.i\, is not zero ; hence 

Cj = 0. 

The vanishing of the coefficient of if'*^' then gives 

0. [.»., 4, + c. b.,.],. + . . . + C, [,j,, ,],_ = ; 

every term after the first vanishes, and [ya.ijg, does not vanish; 
hence 

And so on ; every one of the coefficients G vanishes ; and thus no 
relation of the form 



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124.] OF INTBGEAI^ 389 

Next, there can be no linear relation among the o- membei-s of 
a group. For, in any expression 

SO,,j,„ 
the coefficient of the highest power of log z is of the form 

and this can vanish, only if the coefficients Gs_i are evanescent; 
hence 'ZG^tys.t can vanish, only if the cocfiicients Gs_t a-i'e evan- 
escent. 

Lastly, there can be no linear relation among the members of 
different groups. For let Y{p',z), Y{p",z),... denote the most 
general integrals of the groups belonging to the irreducible roots 
p', p", . . . respectively, of D (p) = 0. Let z describe a contour 
enclosing the origin; then Y(p', s) acquires a factor e'""^, Y(p", z) 
acquires a foctor e^^", and so on. Thus, if there were a relation 

aY{p\s)+0Y(p',z) + ... = Q, 
then 

ae«">' Y(fi', z) + ^e="*" Yip", z)+...^0; 

and similarly, after k descriptions of the contour, 

ae^-ip'' Y {p, z) + /Se=^^"" Y (p", s) + . . . = 0, 

for as many values of the integer k as we please. Now p', p", ... 
are the irreducible roots of D (p) = ; no two of them are equal, 
and no two can differ by an integer. Hence the preceding rela- 
tions can be satisfied, only if 

in other words, no linear relation among the n integrals can exist. 
They therefore form a fundamental system. 



The Equation D(p) = is the Fundamental Equation of 
THE Singularity. 

125. Consider the effect which the description of a closed 
contour, round the origin and lying wholly in the annulus, 
exercises upon this fundamental system. Let 



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390 FUNDAMENTAL EQUATION [125. 

and let y' denote, at the completion of the contour, the value of 
the integral which initially is y. We have 






and so r 
where 



Hence the fun(! a mental equation {Chap, ii) is @ = 0, 



&~e. 


, 


, ■ 


, 








a.i , 


ef-e. 


, . 


., . 








a^i , 




e'-e, . 


, 








, 


, 


, . 


., A, , 


0,0,. 
«'-», 0, . 


, ... 
., , .. 


















■ , , 
.. , 


0,0,. 
0,0,. 












-«, .. 



where a' is the number of integrals in the group belonging to the 
root p of D{p)=0 of multiplicity a' ; a-" is the number in the 
group belonging to the root p" ; and so on. 
Now it was proved that 



Dip). 



n I 



■Kc-p.')"!. 



if 
Henof 



sin (p - p.') ,r = g « - 1' + '.') " (.2"> - e'-' 



^^1"^ - (ZSji'""'"""^''','?/''" '^'■ 



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125.] OF THE SINGULARITY 

Also (§ 117) 

lp„' = in(n-l), 
so that 

g-T'2p„'=,g-in(«-l)Ti^ + 1; 

and therefore 

Dip)- 

As the quantity e""*"^' has no zero for finite values of p, it thus 
appears that, so far as roots are concerned, 2> (p) = and = are 
effectively the same equation, when the relation between p and d 
is taken into account. Also, so far as roots are concerned, il{p)=0 
ia effectively the same as D(p)=0; hence ant/ one of the three 
equations 

= 0, D(p) = Q. n(p) = 0, 

may be used for the determination of 6 and the associated 
quantity p. 

It is known that = is ao equation remaining invariaative 
for all raodiiications of the fundamental system : and, for the form 
of equation adopted in § 114, the term in independent of is 
equal to unity (§ 14), This property in the present cose is verified 
by means of the values of the quantities 8', 0", ...; for 

(_^y(_^Y'... = (-l)''e^'''^''"' = (-lf e"l"-^J^' = (- 1)". 

The remaining coefficients in are known (§ 14) to be the in- 
variarUs of the equation, whatever fundamental system be chosen. 

Replacing by ii (p) for purposes of this discussion, we have 
^ si n {(p - p/) tt] ... sin {(p - p/) 7r| 

^^' sin|(p-p,)7rl...sin|(p-p„),r!- 



I^ow 

where 

so that, as 
and therefore 



sinl(p-p,)»l »-«,■ 



ipr - ip;, 



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392 FUNDAMENTAL [125. 

we have 

^P' (<)-e,)(e-8,)...(0~en) 
^_. .. + (_!)«■ 

Hence, when il(p) is expanded in descending powers of d, the 
term in 8" is unity ; and when it is expanded in ascending powers 
of 6, the term in 0" is likewise unity. 

When the quantities &,, 0^, ..., 6^ are unequal, then Xi(p) can 
be expressed in the form 

On account of the character of il (p), when expanded in ascending 
powers of 8, we have 



so that there are w — 1 independent quantities Jl/,', and these are 
equivalent to the w - 1 invariants. The equation may also be 
expressed in the form 

n{p) = l + t M, cot [{p - p„) tt], 
where 
and therefore 

i #,= 0. 

Corresponding expansions occur in the case when equalities 
occur among the quantities p^, p^, ..., pn- 

126. The integrals, which have been obtained, are valid within 
the annulus represented by R^\s\^R'; the inner circle may 
enclose any number of singularities of the equation, and the outer 
circle may exclude any number of other singularities of the equa- 
tion. But care must be exercised in particular cases. If for 
instance, the only singularity within the inner circle is the origin, 
and the integrals are regular in the vicinity of the origin, then in 
the expression of any integral, such as 



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126.] EQUATION 393 

there can be only a finite number of terms with negative values 
of m : the method, which is baaed upon the supposed existence of 
an unlimited number of such terms, is no longer applicable. If 
the only singularity outside the outer circle is z — <xi, and if the 
integrals are regular in the vicinity of 2 = cc , then in the expres- 
sion of any integral, such as 

there can be only a finite number of terms with positive values of 
m. : the method again ceases to be applicable. 

In 3u<ih cases, the best procedure ia to construct a fundamental 
system which shall include the regular integrals : this ia the 
customary procedure for, e.g., Bessel's equation, the integrals of 
whicli have 3= co for an essential singularity and are regular near 
s = 0. The method, which uses infinite determinants, is best 
reserved for equations which have their integrals non-regular in 
the vicinity of every singularity : it is nugatory when applied to 
Bessel's equation. 



Ex. 1. Consider the equa-tio 



S+S+ b-"- 



It is clear that the point 3 = is an esBential singularity, there being i 
integral regular in its vicinity, when a is different from 0: and that 2=co 
likewise an essential singularity, when y is different from 0. "We ahall aasun 
that both o and y arc non- vanishing quantities. 
Let 

the eqiiation becomes 

dhi (a h a\ 

With the notation of the preceding paragraphs, we have 
4.(p)=p(p-l)-l-6=(p-p,)(p-pj); 



Or,g,=(i, when /t<!'- 1, and when fi> r+l ; 
^,,^=0, when /i<r- 1, and when >i>»-+I. 



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394 EXAMPLES 

Henco the value of n (p) is 



[126. 



..., 0, 


*(p"-"2) 


1 


' *(p-2 


f 


" 


, 


, 0,... 


..., 0, 





0(p-l 


' 


' f(p-i) 


<> 





,0,,.. 


,.., 0, 








' -Hp) 


1 1 


w 





, 0,... 


-, 0, 











^(H^i) 


1 


' *(P + 1) 


, 0,,.. 


..., 0, 





" 


LI 


, 


*(p + 2 


' 


*(7+2)' "'■■■ 



The general investigation ahewa that, wheu p, and p, are unequal (which 
will he assumed), 

Q(p) = l+ifi7rcot{p-pi)n- + J/3^cot{p-pj)jr, 
with the condition 

that is, we have 

Q(p)=l + 7rJ/[coti(p-p,)ff}-Mt{(p-p2),r}], 
where M is itidepeiidcnt of p. 

Taking the determinantal form for Q (p), and espanding according to the 
law established in § 110, we have 

where odd powers of a do not occur because the combinations which they 
multiply all vanish. Also 



, ™+i0{p+m)0(p+m.+l)^(p+?))<^{()+F+l)' 



2 2 



^(p + m)>(p + ™ + l)^(p+p)<^(p+p+l)<fi(p+g)0(^ + 5 + l)' 
Hence we have 



To find jVj, we notice that the only terms in M^, which have p = pi for a 
pole, are those given by m=0, m= - 1, these being 



<^(»^(p+l) 0(p-l)^(p)" 



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126.] 

hence 



<j>{p)={l>-p,){p-p2)\ 



P,-pA<Pipi+V 0(pi-i)J 





Pi-Pa l-ipi-p^y ^^ipi-pi) 




Again, writing 


$(p + m) = ^(p + ra)(>(p + Mi + l), 




we have 


^*=^^*{p + m)*{p+p)- 




Consider 

ij_„*(p+«)}' 

it contains the terms 






ik^J" 




which do not oc 


;cur in i/^,; it cont-vins terras 




which do not oc 


icur in J/, ; and it contains the terras 
1 






^>«+i*{p + '«)*(p+p) 




twice over, onci 


.in the form 




and once iTi the form 

I 




Hence 


»>^+i$(p + m)*(p+^)' 




{?„*(p+7t)) J-A*{p + r>:)i '-.".*(p + ™)*{p4 


-»-n,*^''. 


80 that 






^*=*{1I 


t{p+n)\ ^„J'-. l*{p + «)) -„*{p + ™. 


1 
)1.(p+m + l) 


The first term ■ 


on the right-hand side is 

= i.V3i'^2[cot(p-pi)^^COt(p-p,),r]S; 




the residue of this function for p = p, is 






= -lS\^-,vQat(p^-p^)^ 






-7rC0t{(pj-pj)«-} 





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396 EXAMPLES [126. 

In the second tci'ni on the right-hand side, ttie residue for p=pi can arise only 
for the values n = 0,n= — I; thus it is 



, (l+2p|-2pa)(2 + p,-p,) , (l-2pi + 2pa)(2-Pi + P3 ) 
' {Pi-Pi)Hi+Pi-Pif ^ (Pi-P3)={l-Pi + P2)' 

after reduction. Similarly, from the third term, the residue is 

3-8?> l-8i 

86«(3 + 4i)(p,-p,) Sbm-ih)(p,-p,)- 
Hence 

.r^_w_cot{(pi-p2)^)_ 3-H6-526'^ + lfii3 
* 1662(1-46) " 86^{3 + 46)(l-4i}(pi-p5)' 

after reduction. 



Other coefficients could be calculated in a similar manner : but it is clear 
that even iVg would involve considerable numerical calculations, and it is 
difficult to see how the general term could thus be obtaiued. But the 
method of approsimation may be effective in particular applications. Thiis, 
in Hill's discussion* of the motion of the lunar perigee, the convergence is 
very rapid; and comparatively few terms need be taken in order to obtain 
an approximation of advanced accuracy. When this is the case, the values 
of p' for the integrals arc given by 

Q(p)=0, 

cos2p^=-cos{(p.-p,)^}-2^Jfsin{(pi-ps)^h 
and two irreducible values of p chosen are to be such that 
Pi'+P2'=pi-I-P2 = l- 
The expressions for the integrals are to be obtained. Denoting still by p 
either of the quantities p,' and pj', the relations between the coefficients are 

*oS:7)"'->+°'+.),(,T^'^*'-°- 

and considering in particular the row 0, we know that the constants a are 
propoi'tiona! to the minors of the constituents in that row iu the determinant 
n (p). Thus 

for ail positive and negative values of k : so that, if we take 



imoir already quoted ii 



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126.] 



and our solution h. 



for the effective expression of which, it is sufBcient to find the first minors, as 
the series is known to converge within the annulus. 

I« «rd» to obt.d„ Q torn aO.), we npl,«. ^^^, ^"^-^y ^j^^ 
by zeros ; it will therefore be necessary to do this in the e 
We ttus have 



Q=^l + a-'M^,,+ a'M^, + ..., 



P+J:+S_ 



<p(p)'p(_p + l) 0(p-l)0(p) 

Sinailarly for M^ f from M^ ; and so on. 

In order to obtain [ , j from ii (p), we replace -— — ^ in the — 1 column 
\ - V 9 ip) 

by unity; the quantities 1 and -j-f—~i,\ in that column by zeros; and the 
quantities 1 and -■ . in the line by zero. Wo then easily find 

\-ij 4>{p-i) 



'"'^^(p)<t,{p+i)^<!>{p-i)<i,(py<t.{p-i)<i>{p-2) 



<p{p-l)4<{p-2} 
aud .similai'ly for the others. 
In the same way, we have 



(i)-*iVi)+-*"«*^'..'+-- 



^■f.(p + l)^(p + 2)' 
and so for the others. 



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398 MODES OF CONSTRUCTING [126. 

Lastly, for iiogfttive values of p Icaa than - 1 and for positive values 
gi-eater than -l-l, we have 



{^\==a-^M^,,^-\-a^M„,^-V .. 



where 

M, = Vn , H — — H 

and iu fti the tlei'! 

Alter the remarks made m lalatioa to the foimal development of Q (p) 
it 19 manifest thnt these expressitna for the integrals are mainly useful foi 
apprciimate numerical exparsions they cannot it ptesent lo held ti 
constitute a complete formulation of the mtegiib 

El 2 In his classical memoir alieidj quote 1 H 11 c3naiic.n> tie 



h fli h Kg US erab mailer than «, The mi^inoir 

WW ensaiof ean (jCLOimt bemg takpn of the 

la una, nrgn pLd yPu'ar) and for the numern,il 

approxima ns 

It will bo noticed that the eflectiveness of the method is lai^ly influenced 
by the data as to the smallness of a^, o^, .... when compared with a^. 
Ex. 3. Disciaa the equation in Ex. I, when 6 = J, so that px — H- 
Ex. 4. Given an infinite system of differential equations of the form 

lit 



:J^a^,„:r„, (™^1, 2, ..„«>}, 

where the coefGcients o^^ are regular functions of t within a region \t\ ^ R, 
such that ]«„,„! <S™,d„ in this region, where S^, J„ (for ra, w=l, ...,<») are 
such that the series S,^, + S2^24-...+(S„j1„ + ... converges. Shew that, if a 
set of Constanta e^, Cj, ... be chosen, so that the series 

Cl.d, + C2.d2+...+C„-^„+... 

converges absolutely, then a system of integrals of the equations is uniquely 
determined by the cnndition that a',„ = (^,„, when ( = 0, for all values of ra. 

(von Koch.) 



Other Modes of constructing the Fundamental Equation 
FOR Irregular Integrals. 

127. The preceding nnethod, so far as it is completed, leads 
to the determination of the fundamental equation for a closed 
circuit round the origin, the circuit lying entirely in the annulus; 



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127.J THE FUNDAMENTAL EQUATION 399 

and it leads also to the determination of the integrals. Other 
methods have been proposed by Fuchs*, Hamburger ■(■, Poincar^|, 
and Mittag-Leffler|, some of them referring solely to the con- 
struction of the fundamental equation. But all of them seem less 
direct than the preceding method, due to Hill and von Koch ; and 
they are not less devoid of difficulties in the construction of the 
complete formal expression of the integrals. 

Ex, \. A modification of Hambui^er's method, applied to the equation 



abeady discussed in Ex. 1, § 126, may give some indication of his process. 
Changing the variable from ^ io t, where 



the equation!] i°'' ^ '' 
where 






df 



c=6-i. 

Let X describe a circle round the origin, say of radius unity ; then on the 
completion of the circle, t has increased its value by 2ir. 

Let y=f{^), y=3 (a^) be two linearly independent integrals ; and when x 
describes its circle, let these become [/(if)], \ff (^)], respectively, so tliat 

[/(^)] = «,./(^)+<tisS'W, 

[6'(^)] = %/(^)+«22ffW- 

The fundamental equation for the circuit is 



* CrelU, t. Lxsv (1873), pp. 177—223. 

+ Crelle, t. Lissin (1877), pp. 185—209. In oouneotion with this r 
reference shoulil be made to two papers by Giinther, Crelle, t. cvi (1890), p[ 
336, ib.. t. cvri (1891), pp. 298—318. 

X Acta Math., t. iv (1884), pp. 201—319. In connection with tliis i 
reference should be made to Vogt, Ann. de VEc. Norm., SSr. 3% t. vi (1689), 
pp. 3—71. 

5 Ada Math., t. sv (1891), pp. 1—33. 

II In this form, it is a spcoial case of Hill's equation : nee Es. 2, g 126. 



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400 EXAMPLE [127. 

that is, by Poincare's theorem {§ 14), 

<"^-(l'll+«22)'» + l=0, 

so that Ci, +<i22 is the one invariant for the circuit. 
Let 

The fundamental equation is indepondejit of the choice of the linearly 
independent system, and it is unchanged when any particular selection is 
made. Accordingly, let the integrals be chosen so that 

F(i) = l, F'{t)=0, G{t) = 0, O'W-l, 

when ( = 0; then, using the foregoing equations, we have 

/■(2,).-«,„ ff'(2,).-o„; 
and therefore 

which accordingly gives the value of the invariant, when the values of i''(27r) 
and G' (27r) are known. 

To obtain these, let 



so that a increases from to 1, as ( increases from to Stt. The equation 
becomes 

and this remains unaltered when we change u into 1 - u. Two linearly 
independent integrals, constituting a fundamental system in the vicinity 
of M = 0, are given by 



where a|,=l, Cii=l ; also «„ is the value of 6„ when p=0, and c„ is the value 
of 6„ whet! p=i, the quantities 6„ being given by the equations 

(p + 2)(p + f)6a = !{p+l)2 + 4i^ + 8a}&,-64a, 
and, for values of )t^3, 

()i + p)(!i + p-i)6„={(n + (>-l)'H4c + 8ai6„_i-64(i6,^j + 04o6„_3. 
Similarly, a fundamental system in the vicinity of u~\ is given by 
Z,- 2 «„(!-«)", Z,= S c„(l-«)"H. 



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127.] EXAMPLE 401 

Tho integral F{i), defined by the initial conditions 
/"(() = !, F'{t)^0, 
wlien i=0, is given by 

The integral f?[i), defined by the initial conditions 

i3(0 = 0. ^'[0 = 1. 
when t — 0, is ^b/^n by 

(;(<) = 4F2. 

To obtain expressions for Fi^n), Q' {'in), consider values of u, which 
lie in the vicinity of m=1 and are less than 1, By the ordinary theory of 
linear equations, we have 

First, let «=|, so that 1 — « = ^; then we have 

F{':r) = AF{^)+lBQ{n), a{^) = iCF{w)+DG{n). 

Next, differentiate with regard to a, and then take m=J, 1 — a=J; we have 

F- {^)^ - AF' M-iBG' {t^), 0'{n)=-^CF-{n)-DO'{n). 

Moreover, 

F(()G"(()-F'(0<?(0=constant 

= 1, 

by taking the initial values ; hence 

These rel^ttions give 

A~FM e'M+?"M (?(,). -A 






Hence 



e(2fl--T) = 4r2(2n--T) 

F{2,t~T)=Y^i2n-T) 

.AZ^{r)*BZ,{T), 






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402 EXAMPLES [127. 

Now, when t is tt, the value of u is ^, so that 

and therefore 

■.„+«,.--f'(2,)-e'(!,) 

— L.i!".i. 2-" .:.2— .ioH 

whieh 13 the inyariant of the fmidimeutil equation This gives a fiimal 
expreosion, the onU operatuno requiied lieing m the direct construction 
of i^ and c„, ■ind no one ot thene tperation"; k inverse but the result 
IS less fcuited to numem il ippiiiimation than la the method of infinite 
determinants in the case when a h small 

"We ah^ll return late! (§§ 137 -130) to t difterent diSCusBion of this 
equation. 

Ex. 2. Applj the preoeditig method to Hill's equation 
- -j^=ao+«iC08 2(+aaCos4(+..., 
in the case when %, a^, ... are not small compared with o^. 

Ea;. 3. Discuss, in the same manner as in Ex. 1, the equation 






In particular, obtain espressions for the invariants of the fundamentiil eqiuv- 
tion for z=0. 



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CHAPTEB IX. 

Equations with Uniform Periodic Coefficients. 

128. Am, the equations which hitherto have been considered 
have had uniform functiona of the variable for the coefficients of 
the derivatives ; and the only particular class of uniform functions, 
that has been specially adopted with a view to detailed discussion 
of the properties of the equation, is constituted by those which 
are rational. Many of the properties, however, which have been 
established in the preceding chapters, hold for uniform functions 
whose form, in the vicinity of a singularity, is similar to that of 
a rational function when expressed as a power-series in such a 
vicinity. Among the classes of uniform functions, other than 
rational functions, there are two characterised by a set of specific 
properties ; viz. simply-periodic functions, and doubly-periodic 
functions ; and accordingly, it seems desirable to consider equa- 
tions having coefficients of this type. The present chapter will 
be devoted to the discussion of equations the coefficients in which 
are uniform periodic functions. 



Equations with Simply-periodic Coefficients. 

We begin with the case in which the coefficients have only a 
single period ; and we take the equation in the form 

where p,, ..., p^ are uniform functions of z, are periodic in w, 
and have no essential singularity for finite values of 3. Let a 



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404; EQUATIONS HAVING [128. 

fundamental system of integrals in the domain of any point be 
denoted by 

/.W. /.W /-<4 

which therefore are linearly independent. A change of 2 into z+tn 
leaves the differential equation unaltered : hence 

/.(2 + «), /,(^ + ») /,(^ + «) 

are integrals of the equation. That they are linearly independent, 
and therefore constitute a fundamental system (it may be in a 
new domain), is easily seen ; for 

satisfies the equation for aJl values of s, and by making s pass 
from any position Z+ a to Z without meeting any singularity, the 
integral changes from XcrfriZ + a) to ScrfriZ). If, then, values 
of c could be found such that the equation 

is satisfied identically (and not merely for special zeros of the 
function on the left-hand side), then we should have 

2c,/.(^) = 0, 

also identically. The latter is impossible, because the integrals 
A{z), ■■■,fmi^) constitute a fundamental system; and therefore 
the former is impossible. Thus /, (a + ro), . . , , /,„ {e + oi) constitute 
a fundamental system. 

Suppose now that the domain, in which the original funda- 
mental system exists, and the domaan, in which the deduced 
fundamental system exists, have some region in common that 
is not infinitesimal; and consider the integrals within this 
common region. As fi{z-k-a), ..., /^(•^ + w) are integrals, and 
as/, (e), ...,f,a{z) are a fundamental system, we have equations 
of the form 



/, (^ + ») - o„,/, w + . . . + «.,/. w ) 

where the coefficients a are constants ; their determinant is not 
zero, because the set of integrals on the left-hand side constitutes 
a fundamental system. 



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128.] UNIFORM PERIODIC COEFFICIENIS 405 

Consider any other integral in this region ; it is of the form 

FW = «,/,W+.,/,W + ...+«,A,W, 

where /Ci, «2, ..,, «™ are constants; and so 

ii' (» + «) = 1 <!„«,/, W + I »„«,/, (2) + . .. + I »„«,/, W, 

In order that F(z) may be characterised by the property 

^ constant, the coefficients k must be chosen so 
l<i^«r = ^«p, (;» = 1, 2, ...,m). 



where d i 
that 



a set of n equations linear and homogeneous in the coefficients k ; 
and therefore S must satisfy the equation 

(La , a^-d, ..., a^ 



an equation involving the coeSicients a, and so apparently depend- 
ing ixpon the choice of the fundamental system /i, .,., f^- 

But, as with the corresponding equation for a set of integrals 
near a singularity (| 14), we prove that this equoiion is independent 
of the choice of the fundamental system, so that the coefficients 
of the powers of 9 are invariants. The proof follows the hues 
of Hamburger's proof for the earlier proposition. Let another 
fundamental system g^{z), .... ffmi^)- existing in the region under 
consideration, be such that 



?-(»■ 



")-in».W + .- + f>™9..W. ('■-I.' 



«}. 



the determinant of the coefficients b being different from zero. 
The equation, to be satisfied by the multiplier of F(s), is 



B(S). 



ill - «, 



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406 FUNDAMENTAL EQUATION [128. 

As the integrals / are a fundamental system in the region, in 
which the integrals g exist, we have 

5',(3)-c,,/,(2) + ...+c,^/™(4 (5 = 1, ,..,m), 
where the determinant of the coefficients Cj;, say G, is not zero. 
Thus, as 

hn9, (^) + . . . + K^g^ = gr{^ + ») 

we have 

_l lb„c„f, (2) = l_ l^co., /, (,). 

This homogeneous linear relation among the linearly independent 
integralsy" must be an identity; and therefore 



S hraCst — S C^sffst 



say. Then 






= A(e)C, 



BO that, as C is not z 



we have 
B(0) = Aid), 

and the equation is invariantive. We therefore call it the fun 
mental equation for the period a. 
Let A (z) denote the determinant 

4(.). '?::^^ ^^:^' ^":^ 



1^9, we have 



in— 


da—' ■ ■ 


■■ d.-' 




df, 
dz ' ■ 


df. 
■■ di 


/. 


/. , . 


., /- 



r "fA^)dx 



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28.] 
>that 



FOB THE PERIOD 



A {« + «.) I'^'pil')" 

where we may assume the integration to take place along a path 
that does not approach infinitesimatly near the singularities of pi, 
if any. Now, as pi is a uniform function, simply-periodic in m, it 
is known* thatpi is expressible in the form 

within such a region as encloses the path of integration ; and the 
series is a converging series. But 



ly? 



if the integer a is distinct from zero ; hence 

aw " ■ 

But, substituting in A(s4-w) the expressions for /i (z + w), ..., 
/m(^ + f^} ^rid their derivatives, in terms oi fi{z), ..., /™(s) and 
their derivatives, we have 

A(» + ») 



which is the non-vanishing constant term in A (0) ; and thus 

In particular, when pi is zero, so that the differential equation 
, we have ^„ = ; and then 



contains no term in - 



A(0) = l^ 



K-ir^™- 



129. The generic character of the integi-als depends upon the 
nature of the roots of the fundamental equation, 



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408 ROOTS OF THE [129. 

If the m roots of the fundamental equation are different from 
one another, and if they are denoted hy ff^, 0^, ..., ^m, then a 
fundamental system of integrals exists, such that 

K(^+«)^e,F,(4. (---i '»)• 

Consider any simple root ffr of the equation A ($) = 0. 
Then not all the minors of A {d) of the first order can vanish for 
$ = ds-', hence m — 1 of the equations 

I a,pK,= eKp. (p = h 2, ..., m), 

determine ratios of the m quantities k, and consequently determ- 
ine a function i^r(^) having a multiplier ^,. This holds for each 
of the m different roots: and thus m different functions F{z) are 
determined. 

These m functions are linearly independent of one another. If 
there were an equation 

7i^i (2) + ^,F,(s) + ...-]- j^F^ (^) = 0, 
which is satisfied identically, thea also 

y,F,(s+6y) + y,F,{z+m) + ... + y^F^{z + o,) = Q, 

that is, 

^ijiF, {z) + d^jj\ (s) + ... + e^'i^F^ {z) = 0. 
Similarly, 

^1^71 -fi («) + ^.'7i^. (s) + . .. + 0^'y^F^ (s) = ; 
and so on, up to 

d,"^'yiF,(s)+o,^-'y,F,{0) + ... + e«,-^-'r.>^F>n(s) = o. 

Now the determinant 

W, e,\ 6i, ..., e^'^'l 
does not vanish, because the quantities are unequal ; hence 

so that the constants 7 all vanish. The m functions F therefore 
constitute a fundamental system. 

1.30. Next, let °r be a root of ^(^)=0 of multiplicity ^, 
where ^ > 1. The equations 

^ a,ipKs = 0iCp, (i>= 1, ■■■> m). 



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130.] FUNDAMENTAL EQUATION 409 

are consistent with one another, though not necessarily inde- 
pendent of one another : any m — 1 of them are satisfied hy 
ratios of the quantities k, which are finite and may contain 
arbitrary elements. Giving any particular values to the last, we 
have an integral, say 'I>i {z), defined by means of these quantities : 
it is such that 

and it is a hncar combination of/, (^), ..., f^{z). Taking any one 
of the integrals which occur in the expression of this linear com- 
bination, say /i {z), we modify the fundamental system so as to 
replace f^ (s) hy "!>, (a). Let the equations for the increase of the 
argument by oi in the modified fundamental system be 

/.(^ + t«) = c„*,(^)+c./,(2)4 
then the fundamental equation is 



fc,,„ /,„(.), (r 



'h-6, 







which, owing to its invariantive character, is A {$) = 0, and therefore 
has S for a root of multiplicity /l. Consequently, the equation 



has ^ for a root of multiplicity ^ — t ; and therefore the equations 
(c^-'^)ic^'+c.^ K^' + .-. + o^/cJ =0, 



Cn^f^!' + 0^3*3' + ■ . ■ + (Cmm -^) K 



= 0, 



are consistent with one another, and any m — 2 are satisfied by 
ratios of the quantities «', which are finite and may contain 
arbitrary elements. Giving any particular values to the latter, 
and writing 

«>. W -«■'/. W + «.'/. W + .••+ «..7- W, 

we have 

■D,(» + «) - x,,*, W + a*. W, 



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410 FUNDAMENTAL SYSTEM [130. 

where 

so that Xsi is a constant, which may be zero. The quantity 0^{e) 
is an integral of the differential equation r we use it to replace 
some one of the integrals in its expression, say /^is), in the 
fundamental system, so that the latter then is constituted by 

s.W, i-.W/.W /~W- 

Proceeding similarly from stage to stage, we infer that, 
associated with a root ^ of multiplicity /j. of the fundamental 
equation, there exists a set of ft integrals such that 

0, {^ + w) = V*. (z) + \,,*, (z) + ^*3 (2), 

where the coefficients X are constants. 

Similarly, if the roots of the equation ^ (^) = are %, ...,&„ 
of multiplicities ^u.,, .... n-a respectively, so that /^i + ... + fi,j^ — in, 
the fundamental system can be chosen so that it arranges itself in 
n sets, each set being associated with one root of the fundamental 
equation and having properties of the same nature as the set 
associated with the preceding root of multiplicity &. 

A function, characterised by the property 

is strictly periodic, and sometimes it is said to be periodic of the 
first kind. A function, characterised by the property 

F(, + „).ilF{z), 

where ^ is a constant different from unity, is pseu do -periodic, and 
sometimes it is said to be periodic of the second kind, 9 being 
called its multiplier. A function, characterised by the property 

where X and /j. are constants, is also pseudo- per iodic, and some- 
times it is said to be periodic of the third kind. 



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130,] OF INTEGBALS 411 

With these definitions, the preceding result can be enunciated 
as follows*:— 

A linear differential equation, the coefficients of which are 
simply-periodic in a period co, possesses integrals which are 
periodic of the secmid kind: and the number of such integrals 
is at least as great as the number of distinct roots of the funda- 
mental equation for the period. 
Ex. 1. Prove tliat, if the equation 

iPw , , .civ! , , , „ 

integral which is periodic of the third l^ind with a multiplier 



e"+^, then 

i.i(s+a,)=p,(s)-2X, 

Hence integrate the equation 

shewing that XiB = 4jr^. (Craig.) 

Ex. 2. Shew that, if the coefficients in the equation 

have the form 

p,(.) = 0(.) + ^, 

where i^ and yjr are periodic of the first kind, then the equation certainly 
possesses one integral that is periodic of the third kind. (Craig,) 

131. On the basis of these properties, we can take one step 
towards the analytical expression of the integrals. 

The integral »I>, (s) is a periodic function of the second kind. 
As regards the integral '^^(s), we have 

*,(« + '«) ^A^) ■A' 

' Floquet, Ann. de Vic. Norm., S^r, 2°, t. sii (1883), p. SS. 



that 



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412 PERIODIC INTEGRALS [131. 

SO that the function on the right-hand side is a periodic function 
of the first liind, say t/t (z). Therefore 

where <I'isa (s) is a constant multiple of *i (g), and the constant 
factor may be zero ; and ^^ (z), = i/r (s) *i (z), is a periodic 
function of the second kind, with the same multiplier as *i (a). 



Vs regards 


the integral $3 (3), we 


have 








*, (s + ffl. 












<I>, (2 + 0. 




'.if 




-+w)-i 










-&"*<">■ 


_x,.x. , 

2SW 


. + ^-^ 




2&X. 



we hai 

80 that ^(3) is periodic of the first kind. Hence 

where "t,, (a) = (z) ^^ (a), and therefore is a periodic function of 
the second kind with the same multiplier as "^ij where 'i':si(z) is a 
linear combination of ^^ (s) and ^, {z), and thus is periodic of the 
second kind with the same multiplier as ^i (s) ; and ^si{z) is a 
constant multiple of ^, (s), in which the constant factor, viz. 

may be zero, and certainly is zero if ^-aiz) disappears from (^^{s) 
owing to the vanishing of its constant factor. 

Proceeding in this way stage by stage, we obtain expressions 
for the integrals in succession ; and we find 

<^^ {Z) = •3>„ (S) + Z<^ri {Z) + 3= *rs (3) + . . . + 2'^'*rr (z), 

where 

^ . (- ly X.,r-iXr-.,r- s ■ - X:.,X^i , . 

■^"■W (r-lji^'M-- ''••'' 

80 that it is a constant multiple of *i{^), the constant factor 
being capable of vanishing ; and all the functions 3>^i {z), ^^ (s), 
..., *r,r-i(2) are periodic functions of the second kind with the 



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131.] OF THE SECOND KIND 413 

same multiplier as ^^ (z), and are expressible as linear combina- 
tious of O,, *,i, *3,, ..., <&r-i,i. This holds for the values r = 1, 
2, ...,(^ 

Similarly for any other set of integrals, associated with any 
multiple root of the fundamental equation of the period. 

It may, however, happen that some one of the coefficients 
\,a^i vanishes, so that, for all values of r^s, the term in ^„ (a) 
disappears. The alternative result is that a linear combination of 
the functions ^g{z), ^s-i{z), ..., "I'lia) can be constructed which is 
periodic of the second kind. This linear combination can be used 
to replace ^s(^), and thus may be the initial member of another 
set of integrals in the group associated with the multiplier &. 
The proof of this statement is simple. Assume that \j,j_i vanishes, 
and that no one of the coefficients V,.., for yaluesofr^ a vanishes; 
and construct the linear combination 

choosing the coefficients k so that the term in "J^i disappears and 
that the remaining terms are 

^ («,*, {z) + «,_,*,_, (2) + . . . + «,*, (z)\. 
To satisfy these conditions, we must have 

= «,\sl + K^lXs-1,1 + .■■ + K4V + fs!^3i + «3^, 



= KgXs.t-i + K^^'k^l,,-^. 

Transfer the terms in ics to the left-hand side : the determinant of 
the coefficients k on the remaining right-hand side is 

which by the initial hypothesis does not vanish. Some of the 
coefficients Xj,, Xgj, ,.., Xj,j_a are different from zero, for ^i{z) 
is not a periodic function of the second kind; hence there are 
finite non-zero values for the ratios of «s-i, ■■-, «= to «s- When 
these values are inserted, let 

*.{«)-«.*. W + -.- + «.'i>.Wi 

then 



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414 SETS OF [13L 

so that 'Sfj (s) is periodic of the second kind, with the common 
multiplier ^ ; it can replace <E>s (z) in the fundamental sj^tem, and 
then can, like Oi(^), be the initial member of another set within 
the group of the same type as *i(«), ..., *«-i(s). The statement 
is thus proved. 

132. Any set, such as ^, (s), ..., ^s-iCs) in the preceding 
group of integrals, whether s = /i or be less than /t, can be replaced 
by an equivalent set of simpler form. 

Let the equation be written 



BO that 
Also let 





P, 


SP 




P, 


3-P 


and, 


generally. 


let 




P. 


d'P 






Let the integral of the set containing the highest power of z, 
say g''~', be expressed in the form 

...4(r- l)«0^_, + ^„ 
the binomial factors being inserted for simplicity. Then, as 

F (Z'y}r) = 2-P (f ) + KZ'-'P, (l/r) + «(«- l)2-=Ps W + ■.■, 

we have 
= P(».) 
-^■P(« + ('--l)^'-P.(« + i('--l)Cr-2)i>"P,(« + ... 
+ <r-l)K-'P (,(,,)+ (r - 2) z'-P, (« + ... 

+ «r - 1) (r - 2) {^-P (« + ... 



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132.] INTEGRALS 

which can be aatiefied identically, only if 



The first of these conditions shews that 
is an integral of the equation. The second shews that 
is an integral ; the third that 
is an integral. And generally, if w denote 



^(M-l}!(r-/^)l^ 

...+(7--l)f^^, + 0„ 
w being an integral of the equation, then each of the quantities 
_ _1 a^ 2!(r-3)! 5°w (fi-l)[{r-fj.)]3'-''w 

^"^ r-iar (r-1)! ar^' "'■' (^-1)! S?*^"' ■" 

is an integral of the equation, when ^ is replaced by £ after differ- 
entiation. Accordingly, the group of r integrals in the set are 
linearly equivalent to 

i!^ = .^s + 301, 

'ts = 03 4- 230a + 2^01 . 

M4 = 04 + 3^0, + 32=02 + ^'Vi , 

Ur = ^r + (r~ 1) 30^1 + .. . + (r - 1) 3'-^02 + 2-'<Pu 
and any linear combination of these is an integral of the differ- 
ential equation ; all the quantities which occur in them are 
periodic of the second kind, having the same multiplier. 

Similarly for any other set ; and thus the- vi integrals of the 
equation will he constituted hy sets of r^iVi, ..., r„ integrals of the 
- m, and the system contains 
riodic functions of the second kind. 



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416 group of istegrai£ associated [133. 

Group of Integrals associated with a Multiple Eoot 
OF THE Fundamental Equation of the Period. 

133. These results can aiso be obtained by using the proper- 
ties of the elementary divisors of the quantity A (0), when it is 
expressed in its determinantal form. Let the elementary divisors 
associated with the root S be 

so that, as in § 15, the highest power of ^ — ^i common to all the 
first minors of A {$) is (d — 'Sr)"^, the highest power common to all 
the second minora of A (6) is {d — ^)'^, and so on; and the minors 
of order t (and therefore of degree m — t in the coefficients) of 
A (d) are the earliest in successively increasing orders not to 
vanish simultaneously when 6 = '^. As in the earlier case dis- 
cussed in §1 15, 16, we have 

Proceeding on lines precisely similar to those followed in | 23 
for the arrangement, in sub-groups, of the group of integrals 
aasociated with a multiple root of the fundamental equation 
belonging to the singularity, we obtain a corresponding result in 
the present case, as follows : — 

The group of ft mtegrah aasociated with the root ^ of multipli- 
city n,belonging to the funda/mental equation for the period at, can be 
arranged in r sub-groups, where t is the numh&r of elementary 
divisors of A {d) which are powers of $~^. If the X members of 
any owe of these sub-groups be denoted by gi{z), g^iz), ..., gt.{z), 
these irdegrals of the differential equation satisfy the characteristic 
equations 

J. (^ + »)-as,w •, 

jr. (2 + »)=.*» W+ ft W I 

j.(«+")=aftW+».(») r 



Taking all these sub-groups together, the number of first equations 
which occur in them is equal to the number of the sah-groups, 
that is, the number of the elementary divisors of A {6) connected 



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133.] WITH A MULTIPLE ROOT 417 

with ^ — 9- ; the number of second equations which occur is the same 
as the number of those indices of the elementary divis&rs connected 
with 6 — "^ that are not less than 2 ; the number of third equations 
is the same as the number of those indices that are not less tha/n 3 ; 
and so on, the number of equations in the first sub-group being 

The analogy with the Hamburger sub-groups in Chapter ii is 
complete. 

Corollary. The total number of integrals of the second kind, 
defined as satisfying a relation of the form 
g {! + «) = eg (.), 
where 6 is a constant, is the total number of elementary divisors of 
A(0) associated with all the roots of A(d) = 0; a theorem more 
exact than Floquet's (§ 130). For the total number of such 
integrals, in the group associated with a multiple root of 
A ($) = 0, is equal to the number of elementary divisors of A (ff) 
associated with that root : and the total number of groups is 
equal to the number of distinct roots of A (8) = 0. 

134. Some approach to the analytical expressions of the 
functions, satisfying the equations characteristic of the sub-group, 
can be made, as in § 23. Let 

and introduce a difference -symbol V, such that* 

for any function F; also let 

/X-l\ ^ f\-l\ 



G 



<^) = x^ + (\') ?;..-. + (^ 2 ')rxA-. + ... 



where the functions j(i, ^2, ■-■, %>. are periodic functions of 3, with 
a period ro, and 

V r J rl(\-l-r)': 



* For theae difference -symbols in general, i 
Mat., See. 2", t. x (1882), pp. 10—45. 



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418 GROUP OF INTEGRALS ASSOCIATED [134. 

Then if we take 

for a!! values of m, we have 

holding for all values of n. These are the characteristic equations 
of the sub-gi^oup ; and we therefore can write 

with the above notations, for n = 0, 1, . . X — 1 

These \ integrals are a linearly independent ''ei out of the 
fundamental system; the system will remain fundunental if 
ffi! 9i> ---.^A *re replaced by X other functions Imeailj equivalent 
to them and linearly independent of one inothe-r This modifica- 
tion can he effected in the same way as the corresponding modifi- 
cation was effected in | 24, viz. by introducing a set of functions, 
associated with G and defined by the relations 

the functions '^^ being periodic functions of z, with tho period w. 
Constructing the expressions Vff, V^G, ..., V'-^G, we find 



V^-^G = cx_,,,Gi, 

where the constants c are no n- vanishing numbers, the exact values 
of which are not needed for the present purpose. 



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134.] WITH A MULTIPLE ROOT 419 

It follows, from the last of the equations, that G^ is a constant 
multiple of V*~'{?, and therefore that ^" G, is a constant multiple 
of ffi (s) ; we replace y, (s) in tlie fundamental system by ^ G^. 

It follows, from the last two of the equations, that G^ is a linear 

combination of V—^G and V'~^G, and therefore that S^"(?s is a 
linear combination of g^ (z) and g-i (s). As g^ {z) has been replaced 

in the fundamental system, we now replace gi{z) by ^"Gj; and 
the system remains fundamental. 

And so on, for the integrals in succession. Proceeding thus, 
we obtain X, integrals of the form 

?ri9.(2), ^" (?,(«), ...,S-iS;^(s). 
Further, these integrals are linearly independent, and so they are 
linearly eqiiivalent to ^,(3), ^2(2), ...,gx(z). For if any relation, 
linear and homogeneous among these quantities, were to exist 
with non-vanishing coefficients, we should, on substitution for 
Gu G„ ..,, (?;vin terms of G',VG,V=G',.. .,V*-G', obtain a relation, 
linear and homogeneous among the quantities gi{z), ..., g^i^) 
with non-vanishing coefficients. Such a relation does not exist. 
Accordingly, the X integrals 

can be taken as constituting the required sub-group of integrals. 

We now are in a position to enunciate the following result, 
defining the group of integrals associated with a multiple root ^ 
of the fundamental equation of the period : — 

When a root ^ 0/ the fundamental equation A (0} = O is of 
midtiplidty fi, there is a group of fj, integrals associated with that 
root; the group can be arranged in a number of sub-groups, their 
number being equal to the number of elementary divisors of A {$) 
which are powers of "it — 6 ; the number of integrals in the fi/rst 
sub-group is equal to the number of those elementary divisors ; the 
nwmber in the second sub-group is eqtial to the number of the 
exponents of those divisors which are equal to or greater than 2 ; 
the number in the third sub-group is equal to the number of the 

27—2 



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420 GROUP OF INTEGRALS [134. 

eicpan&ixts of those divisors which are equal to or greater than 3 ; 
and a suh-grmip, which contains X integrals, is equivalent to the X 
linearly independent quantities 

where 

ftW-X.+ (''7')x-.f+(''2')x-.f + - 

for r = ], 2, ..., X-' the quantities %,, ,.,, y^^ are periodic functions 
of z, but they are not necessarily uniform : f denotes — , and 
fr-l\_ (r-1) ! _ 

Note. By taking ;:^„ = q>~"0„, for m = 1, ..., \, and wiiting 

the integrals become 

e. (^) =4>.+ ('■ ^ ^)>._,^ + r ^ ^) ^^^^ + . , . 

the functions i^ having the same character as the functions )(. 



135. There is a theorem of the nature of a converse to the 
foregoing proposition, which is analogous to Fuchs's theorem 
proved in ^ 25 — 28. The theorem, which manifestly is important 
as regards the reducibility of a given equation, is as follows :— 

If an expression for a quantity u is given in the form 

w = ^" \4,n + ^„_i? + 4>„^,K' + ■ - + W-" + W-'l 

where & is a constant, all the functions cp-^, ..., 0„ are periodic in a>, 
and ^ denotes - , then u satisfies a homogeneous linear differential 



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135.] CONSTRUCTION OF UNIFORM INTEORALS 421 

equation of order n, the coeffidents of which are uniform periodic 
functions of z, Iiaviiiff the period a; -moreover, 

are integrals of the same equation and, taken together with u, they 
constitute a f undo/mental system for the equation of order n. 

The course of the proof is so similar bo the proof of the corre- 
sponding theorem as established in §§ 26 — 28 that it need not be 
set out here*. It can be divided into three sections ; in the first, 

it is proved that — , ..., ■ -^ satisfy such an equation, if u 

satisfies it ; in the second, it is proved that these must form a 
fiindamental system, for no homogeneous linear relation with non- 
evanescent coefficients can exist among them ; in the third, it is 
shewn that the linear equation, which has these quantities for its 
fundamental system, has uniform periodic functions of z with 
period w for its coefRcients. The details of the proof are left to 
the student. 



Mode of obtaining Integrals that are Uniform. 

136. The further determination of the analytical expressions 
of the integrals, on the basis of the properties already established, 
is not possible in the general case. Thus the functions )(i, ..., %a> 
occurring in the sub-group specially considered in § 134, are 
periodic functions of the second kind with a multiplier ^. If we 
take new functions ■^i{2), ..., "^/.(f), such that 

these new functions are periodic of the first kind. But further 
properties of the functions must be given if there is to be any 
further determination of their form. 

When we limit ourselves to the consideration of those equations 
whose integrals are uniform functions, (criteria are determined 
* Some of the analysis of g 133 ig useful in establishinf; the theorem. 



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422 EQUATIONS HAVING [136. 

independently by considering the integrals in the vicinity of the 
singularities), some further progress can be made ; bub, of course, 
the assumption that the integrals are of this character must be 
justified by appro|)riate limitations upon the forms of the coeffi- 
cients Pi, ..., pm in the original dift'erenbial equation. In such 
cases, every quantity such as -^^C-^) i^ a uniform simply-periodie 
function of the first kind; it can therefore* be expressed in the 
form of a Laurent- Fourier series such as 

Such a, form of expression does not lead, however, towards the 
determination of the criteria for securing such a result or any 
other result of a corresponding kind for any other assumption. In 
particular examples, we adopt a different method of practical 
procedure. 

In order to determine some of the functional properties of the 
integrals, it frequently is expedient to change the variable so that, 
if possible, the transformed equation belongs to one or other of 
the classes of equations considered in preceding chapters. 

Thus if the coefficients^!, ...,pm, which are uniform periodic 



then, introducing a new variable (, where 

we obtain a linear equation, the coefficients of which are rational 
functions of t. Some characteristic properties of the integrals 
of the equation in the latter form can be obtained by earlier 
processes ; it may even be possible to determine the fundamental 
system of integrals. 

The preceding transformation is, however, not the only one 
that can be used with advantage ; and it often happens that 
the special form of a particular equation suggests a special 
transformation which is effective. In pai'ticular, if the coeffi- 
cients in the equation are alternately odd and even functions, 

' T. F,, % 112, 



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SJMPLY-PERIODIC COEFFICIENTS 



such that pi,pi,Pi, -.. are odd, and p2,pi,pe, ■■■ are even, then 
wc may take 



as a new independent variable r it is easy to prove that the 
transformed equation has uniform functions of ( for its coeffi- 
cients. Also, some indication is occasionally given as to a choice 
between these two transformations ; for example, if an irreducible 
pole of the original equation is a = 0, we should choose 

( = sin — 

aa the transformation, and consider the integrals in the vicinity of 
t = 0; whereas the other would be chosen, if an irreducible pole of 
the equation is s = ^oi. 

Another transformation, that sometimes can prove effective, is 









any uniform function of 3, periodic in o>, can be expressed as a 
uniform function of t ; and the differential equation is transformed 
into one which has uniform functions of t for its coefficients. 

Ex. 1. Couaider the equation 

J'+2.:^cot.+ (6 + .oot..)„.0, 

where a, b, c are constants. Writing 

we have the equation 



whei'e 



and the oquatio 



+ (/3+T-cot2s)», = O, 



The indicial equation for t—0 is 

p(p-l) + y=0. 
If v=Saj''*'' 



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424 EXAMPLES [136. 

satisfies the equation, we have 

n2+2n(p--2) + 4--0-3p 

'• if+M^ir^) *■"'■ 

The form of a„, in terms of o^-a, shews that the series for y converges for 
values of ! ^ 1. If the two roots of the indicia! equation arc p^ and p^, and 
f{i,Pi),f{t, Pa) ho the two values of y, the primitive of the original equation is 

,>— in-. |J/(siii 4 p,)+il/(«n ,, a)}. 

Ex. 2. Consider the equation 



we find the transformed equation for i to be 

which is Legendre's equation and so its primitive is 

Ex. 3. Obtain int^rals of the equations 

-j-5 +-J- cot!--wcoseo^3=0; 
d^ dz 

d^ die a a a —ft. 

dz^ da ~ ' 



(i) 
(ii); 

(-)S-(s-nr^-OSHovk-.-i.^)^- 

Ex. 4. One integral, /(^), of the equation 
4(i!-,i„.)5 + !.(3.m. + 2c„..^«)* + (5-3co..-,m.),.0 
the relation 



find the general solution. (Math. Tripos, Part it, 1896.) 

Ex. S. Shew that the equation 

has an integral 

where S[ has an appropriate constant value; and obtain the primitive. 

(M. Elliott.) 



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136.] liapounoff's investigation 

Ex. 6. Obtain an integral of tlie equation 



where A is a constant, in the forin 

^_sin(?-gi) sin(a-33) ^(cots^+cots,) 

where 2, and e^ are appropriate determinate constants ; and obtain the 
primitive. (M. Elliott) 

E.V. 7. Integrate the equation 

where w is an mt«ger, and A is a constant. (M. Elliott.) 

137. A somewhat different form of the theory is developed 
by Liapounoff*, whose investigation deals with a more general 
equation, given by 

where /i is a parameter, and p (s) is a uniform periodic function of 
period €0. 

Let f(z) and ^ (e) he two integrals of the differential equation, 
respectively determined by the initial conditions 

/(0).1| *(0).01 

/'(o)=or f(o)=ir 

Then we have relations of the form 

and the equation for determining the multipliers is 

(n-n)(n-8)-^7 = 0, 
that is, 

a= - (a + 5) fi + 1 = 0, 

as in § 127, Ex. 1. Clearly, we have 

* Comptes Hendus, t. cxxm (lS9e), pp. 1248—1252; ih., t. cxSTiii (1899), 
pp. 910—913, 1085-1083. 



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426 liapounoff's [137. 

BO that, if we write 



the equation is 
Writing 



;"-24n + 1.0. 
= A+(A'-l)t, 



and assuming that A'—l does not vanish, we obtain two integrals 
in the form 

where Fj (s), F^ {z) are functions of z, periodic in to ; and thus the 
complete priniitive of the equation can be obtained. The actual 
expressions for J'i(s) and F^{z) can fee constructed as in the 
preceding sections ; and the value of p depends upon that of A . 
When ^ = 0, the primitive of the original equation is 

shewing that the equation for dotermining the multipliers is 

(li- 1)^ = 0; 
and then A = \. Hence, when /a is not zero, and when A is 
expanded in powers of /*, it is inferred that A is oi the form 

^ = 1 — fi,Ai + f>?A^ — iJ?A,^ + .... 

When .^1, A3, ^3, ... are known, the two values of il, which 
satisfy the equation 

n'^-24 11 + 1=0, 

can be regarded as known, and the primitive of the dift'erential 
equation can be obtained. 

For the purpose of obtaining the value of A, which is 

4 = i(/W + f(»)l. 

where the integrals f{z) and <^ {s) are deiined by the initial 
conditions, we assume both f{z) and {z) expanded in powers of 

f{z) = Wo + fiM-i + ij?u.i + . . . ; 
then, in order that it may satisfy the equation 



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137.] 






EQUATIONS 


we have 








and so on 


, From the first, we have 




Ik 


= a, 


, + *.«; 


from the 


second, 


we 


have 



from the third, we have 



= ai + b,z — j rfy I u„(a:)p{x)da:; 

ve have 

= C!a + 632 - 1 dT/j «i (x) p («) da: ; 



and so on. Now 

/(0)=1, /'(0) = 0; 
accordingly, 

a^ + /j.a, + f>?-a.i + . . . = 1, 

ha + iih^ + f.% + ... =0. 

Taking account of the fact that /j, is parametric, we have 

«„ = !, a, = for s = l, "2, ..., ?i, = for s-0, 1,2, , 

and thus we have 

M„=l, 

Wi = -j (^^f p(a:)da:, 
and so on. The value oif(z) is given by 

/(s) = 1 + ^Mi + /t'Ms + . . , . 

Similarly for (2), which is determined by the conditions 

<^(0) = 0, .^'{0) = 1; 
its value is given by 

<j>{s) = e + fiVi + /iH'a + . . . , 



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428 liapounoff's [IS?, 

where 

Vi = -j dyj xp{a;)dx, 

V, = -\yy\yia:)v,{x)dx. 
and so on. 

We require the quantities /(w) and '^'(oi): let thorn be 
denoted by 

where 

[/j = — I dy\ p{x) dx, 

T, = — I cnp{(£) dx, 
and so for the others. Substituting the value of ^ in the form 

-4 = 1 -/tjl, + /iMi. — ..., 
we have 

= 1 dy\ p{x)dx+ i xp(x)dx 

= j %j ^(«)'^ic + j yp(y)dy. 
But 

^ 1^ f^^i? (a') (^a^l = f V (*) -^^ + yp iy) ; 

integrating between the limits and a, we have 

Next, we have 

iA, = -11.-r, 

To transform these definite integrals, we write 

j'"j,W<fa-PW, P(»)=n. 



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137.] METHOD 429 

so that 

uAi!) = -t'dtf'p(6)de = -r P{t}dt, 

v,(a:) = -!'dtj'Sii{e)de 

~-rtP{t)dt+rdtl'p{S)de 

= {'dti' {F{lf)-P(t)]ie. 
We have 

J- 1"! {s)j^p(s) d!/\ - «. (y)p to) -P'(s); 

therefore 

and thus the first integi-al in tlie expression for 2A^ is equal to 

j'dyj'{F{g}-F(a:)]P(^)d^. 
Similarly, we have 

therefore 

= -nj'\P(!,)% + n|^"<!yJ'p(a.)(i» 
+ j''yP-(y)*-JJ<ij/'P&)PW<i«:. 
The first and third terms on the right-hand side together are 
--JJ(n-P(y)|P(y)t,<iy 

= -/J.iy/V-P(y)lP(y)<fc. 



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430 liapoukoff's equations [137. 

ao that 

I" V, (:v)p (.^) <^ = - /^" d^j' [ii'P (y)} \P iy) - P (.^)} dx, 

which gives a traasformation of the second integral in ^A^. 
Combining the results of transformation for the two integrals in 
2-^2 , we have 

HA, = 1^°' d>j r {il-P{y) + P {x)} {P (y) - P (^)i dx. 

Similarly, it may be proved that the value of ^A^ is 

j'Ji/J%/^(n-PW + PW!|PW-P(y)HP(s)-PWi<fc, 

and so on i so that the value of A, and therefore the value of P, is 
known. 

The investigation is continued by Liapounoff, especially for 
the purpose of discussing the values of fi. which satisfy the 
equation 

^=-1 = 0; 

aad the results appear to be of importance in the discussion of 
the stability of motion. The reader is referred to the notes by 
Liapounoff already cited (p. 425, note) ; other references to more 
detailed investigations are there given. 

Esc. 1. Establish by induction, or otherwise, the general law for the 
coefficients A, viz. 

2.t„= <h!,\ rf%... I @rf«„, 

where 

©.(ii-P(i,)+P(«.,)){fW--PW]fi'W--P(».))... If ('.-J-i" (».)>■ 

Ex. 2. Shew that, if the peiiodic function p(s;) always is positive, then 
all the coefficients A are positive; and prove that 



Hence shew that, when p (x) is positive and satisfies the inequality 



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137.] EQUATION OF THE ELLIPTIC CYLINDER 431 

Ex. 3. Prove that, if the periodic function p {m) be real and odd, so that 
the series for A contains only even powers of /t, then 

A^= 4 I °'(fej r'dx^ fds;^ i"' [P^- P^f {P.^~ P^f dx^, 
and MO on, where P,. denotes i'i^r)- Prove also that, if 

tho constant a being determined so that I PciK=0, and if 
"" P''dx:^i, 



'I? 



then ji^<I. 

Ex. 4. Discnss the values of )i which are roots of the equation 

(All these results are due to Liapounoff.) 

Discussion of the Equation of the Elliptic Gylindbe. 

138. One of the most important equations of the ciass, which 
has been considered in § 137, is the equation 



commonly called the equation of the elliptic cylinder; it is of 
frequent occurrence in mathematical physics and astronomical 
dynamics. It forms the subject of many investigations* It is 
known (g 55) to be a transformation of the limiting form of an 
equation of Fuchsian type. Moreover, it has already (| 127, Ex. 1) 
been partially discussed in connection with another equation and 
for another purpose. In this place, it will be brought into relation 
with the preceding general theory. 

Let new independent variables u ov v h& introduced, such 
that 



* Heine, Handbuch der Kugelfanctianen, 1. 1, pp. 404- — 415; Lindeniann, Math. 
J)iji.,t. XXI! (1883), pp. 117—133; Tisaerand, Mecanique Cilcste,t. in, oh. i, at the 
end of which other referenoes are given. 



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432 EQUATION OF THE [138. 

The equation becomes 

.d^w , ,, „ , dw , , ^ , 

"* *^ ~ "^ rf^? "^ *^^ ^ ^""^ rf^ + ^^"^ ~ ' + ^'^''^ "^ = "^' 

when u is the independent variable ; and it becomes 

, dhi) , ,, „ , dw , , ^ , 

when V is the independent variable. Accordingly, if 

is an integral of the equation, another integral is provided by 

»=/(., -c). 

The indicial equation for m = is 

p{/'-^) = 0; 
if 

w = ta^vP+i' 

be the integral, the scale of relation between the coefficients ap is 

(p + p)(p + p-^)S=KP + p-l)°-i(a-c))ap-i-^cap_a, 
with the relations 

«o= 1, 

When p = 0, let fflp = ^ {p, c) ; when p = ^, let aj, = ?y (^, c). Then 
two integrals of the equation are 

X,- i u^e{p, c), a^i= i ?!P+sa(^, c), 

M'ith the convention 

t)(0, c) = l=Sy(0, c). 

It is clear that, when z is Jtt, so that m is 0, 



..=,, J-=o, 


' ' dz 


-1. 


r, as the equation in w i 


s satisfied by » 


„ and 


dx, dx 


C 





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138.] ELLIFHC CYLINDER 433 

But (^M = - 2 Bin s COS ads = - 2 [u (1 - u)\idz, so that 

' ds " dz 
When 2 = ^7r, the left-hand side is equal to 1 : hence 

and therefore, for all values of z, we have 

dxa dxi __ 
^ dz " dz 

Two other integrals of the equation are given by 

they are such that, when z is 0, and therefore v is 0, 

■^° dz ^ dz 

and, for all values of s, 

dy-i dtig 

■^^ dz ■^ ds 

Now when z is real, both m and v are real and lie between 
and 1 ; and, in particular, when 3 = ^7r, then « = •«= |. For such 
values, x„, x,, y^, y-,, coexist ; and so we have relations of the form 

where a, (S, -/, S ai-e constants. Hence 

y. (i) = ™. G) + /3a>, (i), -</.'(« = «; (i) + Hi'l (J), 

where 






and so for the others. Hence 

« = -j.(i)<a)-y.'(i)«.(i). 
/3= y.^^/ffi + y.' (»■».&)■ 



Similarly 



7 = -</.(!)<(» -y.'(i)''.(i)l 

S= J(.tt)»=.'(i)+!/.'(l)«.(J)l' 



F. IV. 



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434 EQUATION Of THE [138. 

and it is easy to verify that 

Moreover, we have 

■*» = %!. - /3i/il 
^1 = - 7^0 + «yJ ■ 

139. The integrals x^ and a;, are valid in the domain of w = ; 
tho integrals y„ and y^ are valid in the domain of v = 0, that is, of 
u= 1. Lindemann* proceeds, as follows, to obtain uniform inte- 
grals valid over the whole of the finite part of the plane. 

After a small closed circuit of u round its origin, x^ returns 
to its initial value and «, changes its sign ; hence y^ becomes 
aic„— /3«], and j/i becomes yxn — hxt. After a small closed circuit 
of u round the point 1, the integral i/o returns to its initial value 
and y, changes its sign. Consider a quantity ij, where 

jj = Ay^ + %l^ 
as a function of u. It remains unchanged when u moves round 
the point 1. Its two values in the vicinity of it = are 

{A-x' + £7=) x^ + {A^ + £8=) x^ + 2 {Ai0 + B7S) x,x^, 

{Aa^ + Brf) x," + {A^^ + BS') x/ - 2 (Aa^ + SyS) x,x„ 

which are the same if 

Aa^ + By& = l}: 

henee the function is uniform in the vicinity of u = if this 
condition is satisfied, that is, the function is uniform over the 
whole plane. 

The condition is satisfied if we take 

and then 

n = tt^y," - ySyo". 

Moreover, in the region of existence common to y^, y,, x„. a;,, we 

'^^yi' — 7^3/11° = 0Sxi^ — ayxa\ 

Hence defining the function 71 in the domain of 3= by its value 

in terms of y^ and j/i , and defining it in the domain of 2 = 1 by its 

value in terms oi x^ and a-'i, we have a function 

^ = F{u) = F(>Ms's)=^ (i), 

• Math. Ann., t. xxii (1883), pp. 117—123. 



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139.] ELLIPTIC CYLINDER 435 

say, which is regular in the vicinity of m = 0, regular in the vicinity 
of w = 1, and therefore is regular over the whole iinite part of the 
z-plane. Now let 

F,-y,(a/3)t + s.(7S)ll 

7.-!,,(.«)l-S.(,S)tf' 
then 

= -2(=i/37S)l. 
Also 

and therefore 

l'«t-+5^.?^" = <E>'(^)- 
ds dz 

Hence 

Y, dz ^ ^{z) <E>(2) ' 
1 dY,_,^'(z) (a^78)'. 






and therefore 

where 

These integrals of the original differential equation are valid over 
the whole of the finite part of the plane. Accordingly, we may 
take two integrals 

M {-- 

(?,(2) = {a>(2)j4e' -'*w 

as integrals, which are valid over the plane and have z—'X for 
their sole essential singularity. We now proceed to shew that 
they are uniform over the plane. 

Substituting in the original differential equation, we have 
{a + c cos Is) <^-' - 1^'-' + ^4><l>" + M= - ; 



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436 lindemann's [139. 

so that, as if in general is not zero, any root of 'I' = is a simple 
root. Let k denote such a root : then 

Now let z describe a simple closed contour, including k and no 
other root of $ = 0, and passing through no root of "5 = 0. Then, 
at the end of the contour, |*^(2))S has changed its sign. As for 
the exponential factors in G{z) and 0,(z), they are multiplied by 



respectively, the integral being taken round the contour, that 
is, they are multiplied by 



^m!'}-^, 



{k) 



that is, by —1. Thus G{z) and G^(s) are unaffected by the 
contour; they are therefore uniform in the vicinity. Moreover, 
in the immediate vicinity of k, we have 

^(0) = (z-k}^'{k) + ..., 
so that 

GA^)={<^'(k)]^(^-k)e-^^'-''^Q(z-k), 

so that A: is a simple root for one of the integrals and it is not a 
root for the other. Similarly, in the vicinity of any other root of 
^ = ; hence G and G, are uniform over the whole plane. 

Now take any path from z to s + tt, for tt is the period for 
the original equation. We have 

where F is uniform ; hence 

i,(,+ ^)—H^), {*{^ + ^))4 = (-l)'jO(2)jS, 

where r is or 1, depending upon the path from to tt. The 
effect upon the exponential factor of G (e) is to multiply it by 






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139.] METHOD 437 

Wo know that ^{s) is regular over the whole plane, that it is 
periodic in ir, and that it has only simple roots ; hence, taking a 
path between z and a + tt, that nowhei-e is near a root, we can 
1 



valid everywhere in the range of integration. Then 

and, consequently, if 

then 

Similarly 

Hence G and G^ are the two periodic functions of the second kind, 
which are integrals of the original equation* ; and they have been 
proved to be uniform functions, regular everywhere in the iinite 
part of the plane. 

Es. Shew that the equation 

has two particular mtegrals the product of which is a, single-vahied tnius- 
cendental function. F{z) ; acd shew that the integraLs are 

"'-'"•""'■"-■[-"/ (.(i^'i^w ]- 

where C is a determinate constant. In what circumatatices are these two 
particular integrals coincident ^ (Math. Tripos, Part ii, 1898.) 

liO. The multipliers /m and — are thus the roots of the 

equation 

D.^- in + 1=0, 

' This ineluaion of Lindemann'a apeeial result within the general theory is dua 
to Stieltjfls, AetT. Naehr., t cii (1884), pp. 147, 148. 



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438 INVARIANT OP THE EQUATION [140. 

where the invariant / of the period o> is 

Another expression for this invariant, consequently leading to 
another mode of obtaining these multipliers, has already been 
given in Ex. 1, § 127. Both processes are dependent upon the 
determination of simple special solutions of the original differential 
equation. 

Another method of proceeding is as follows. Let 

so that 

so that, if 

(?(£)=e*''©(3), 

then, as 6(z) is a uniform function of a, regular over the whole 
plane, (s) is a uniform periodic function of the first kind, regular 
over the whole plane ; and ir is the period. Hence we have 

and therefore 

Now in the vicinity of a- = 0, the integral y^ is even and yi is odd ; 
hence G(z) contains both odd and even parts. The form of the 
differential equation shews that, if /(z) is an integral, then f{—z} 
also is an integral ; hence, as (a) exists over the finite part of 
the plane, G(—z) also is an integral. Henco, taking 

where a is an arbitrary constant, it follows that Il{z) is an integral 
of the original equation, which exists for all finite values of z. 
Substituting in the differential equation, and noting that 

cos 2s cos [(2m + h)z + a\ 

= ^ cos {(2n -2 + h)s + ix} + ^cos [(2« + 2 + h)s + a.\, 
we have 

"s" i{a - (2n + hy\ K„ + ^v (>c„^, + Kn+,)'] cos {(2n + h)s + u}^0. 



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140.] OF THE ELLIPTIC CYLINDEK 439 

as an equation which must be identically satisfied ; hence 

{a - (2k + hf] «„ + Jc (k„_, + K„+0 = 0, 

for all values of n from -co to + a^ . 

The mode of dealing with this infinite set of equations by 
means of infinite determinants has been indicated in a preceding 
chapter, and much of the analysis of the first example in § 126 
is directly applicable here : so we shall not further discuss this 
mode of obtaining h and the ratios of the coefficients «. There is, 
however, another method of obtaining these quantities: it is due 
to Lindstedt* and is specially adapted to the differential equation 
under consideration, for purposes of approximation when c is con- 
veniently small. Writing 

«,. = 2 (2ii + hy- - 2«, 



?' 


C 


1- 

c 




<Xn 


a„a„+, a„+,a„+<j 


1- 


1- I- 



Owing to the form of — for increasing values of r, it is easy to 
prove that this infinite continued fraction converges, for all values 
of n. We therefore have 



Similarly 



K„ 1 - 1 - 1 - 


inf 


c t= c'^ 




«_„ o_„a_„_i a_„_ia_n-. 


'■ ... ad inf.. 


,, 1- 1- 1- 


1. dn I'Acad. Si Pitersbourg, t. xxx 


1 (18B3), No. 4. 



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440 lindstedt's [140, 

which is a converging continued fraction ; and, in particular, 
c c" d" 

/Co 1 — 1 — 1 - ■ ■ ■ 

But, from the fundamental difference -equation, 

therefore 

g2 g3 (.2 p2 p! C^ 

1 _ °^»"i °i^ C2W3 , °ut(-i 0.^1 a_j a-;0:-; 
1 - 1-- l-"""^ 1- 1- 1- ■■■' 
a transcendental equation to determine h, which of course is 
equivalent to the corresponding equation arising out of the 
vanishing of the infinite determinant 1) (p). 

Denoting the first continued fraction by - and the second by 
^ , so that these values may be regarded as coiivergents of infinite 
order, we easily find 

r=a «=r+a (-S+2 OrOr+i a,a^i Ojat+i J 

o = 1 - i -^ + i i -^ ''' ■ 

-si i — ' ^^L.+ ...; 

r=lj=r+2 (=8+2 ar«r+I aa«s+i OfOi+i 

the values of ^' and q' are derivable from the expressions in p 
and q respectively, by changing a^ into «_„ (for all values of /t) 
wherever a^ occurs. 

The equation manifestly lends itself easily to successive ap- 
proximations. Thus, if we neglect C and higher powers, we have 

which, to this order of approximation, gives 

The calculation of the coefficients can similarly be effected. 



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140.] 



METHOD 

Pi-ove that, up to sixth powers of c inclusive, 



< -i , 1 <^ 



1024c[2(l-a)3(4-a) 
105a'-H55a'+3815w=-4705aHl653(t- 



(In astronomical applications, a it 
pared with a.) 

Ex. 2. Taking k^=\^ and writing 
prove that, up to e^ iuclusive, 



1(1-.)' (4- 
mally not ai 



ger, and c is small com- 
(Poincar^, Tiaserand.) 



U+? 10243(l+j)9(2 + s')(l-g)r 






^\\-q 1024j(I-9)'(2-y)(l+2Jj 
(|c)=f coa(Z+fe) , _oo8(£-4^)_l 

■^ 2! l(l + j)(2+j)'^(l-y)(2-3)J 
(i«:^f c oa(g+6. ) , eoa{Z-_6^ 



3! \(l + 5)(2+3)(3 + j) 



whore y^^ 



"(l-j)(2-S)(3-y)J' 

(Poincar^, Tisserand.) 



Ex. 3. In the investigation of § 138, the quantity if is supposed to be 
different from zero. When M is zero, the integrals Q{z) and Q^if) aro 
effectively the same ; and neither of them is uniform, so that the remainder 
of the investigation does not apply. 

Discuss the case when j¥=0. (Heine.) 



Equations i 



[ Uniform Doubly- periodic Coefficients, 



141. We proceed now to the consideration of linear equations, 
the coefficients in which are uniform doubly-periodic functions of 
the independent variable. Let the equation be 



= 0, 



where p^, ..., p,„ are uniform functions of z, which have no 
essential singularity in the finite part of the plane and are 
doubly- periodic in periods to and w', such that the ratio of oi' to w 



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442 DOUBLY-PERIODIC [141. 

is not purely real. A fundamental system of integrals exists in 
the domain of any finite value of z, and may be denoted by 

/.W. /.(') /-«, 

which accordingly are linearly independent of one another. 

The differential equation is unaltered when 3 + lo is wiitteii 
for 2 : hence 

/,(^+«), /,(^+„) /„<,+») 

are integrals of the equation and, as in § 128, they constitute 
a fundamental system of integrals. Similarly, as the differential 
equation is unaltered when a + to' is written for 3, 

/.(«+»'), /.(^+»') /■(^+»') 

constitute a fundamental system of integrals. Choosing therefore 
a region common to the domains of these three fundamental 
systems (a choice that always can be made because the singu- 
larities of the integrals are isolated points, finite in number 
within any limited portion of the plane), we have relations of the 
form 

/, (« + «) = o„/,(^) + ...+o„/„W| 



/. (^ + " ) - «»l/l («) + ...+ 0,.»/,(2) 

and 

/. (^ + .-)_4,. /.(.)+. .. + i,./„W 

/,. (2 + »■) . J,,./, W + . . . + (.„/. W ) 
valid within the region chosen. The coefficients a are constant, 
and their determinant is not zero ; the coefficients b also are 
constant, and their determinant also is not zero. The two sets 
of relations may be represented in the form 

/(^ + »)-s/w, /(^ + »)=.S7W, 

where S and S' denote the linesir substitutions in the relations. 

The coefficients in the two substitutions are not entirely 
independent of one another. We manifestly have 

/,((^ + .) + »-l-/,l(^ + »■) + -!, 

for all values of r. The symbolic expression of this property is 

/((. + ») + ..■) = S'f(z + „ ) = S'iv w, 
/!(. + «■) + « I = s/(« + »')-ss7W, 



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141.] COEFFICIENTS 443 

80 that 

or the linear subsfcitutions are interchangeable. The explicit 
expression of relations between the constants is obtainable from 
the equation 

br:A (^ + «) + &,./. (^ + a>) + . . . + 1™/,„ (^ + o>) 

= a.,/, (z + 0.') +«„/, (^ + »')+... + a™/« (^ + o>'), 
by substituting for the functions / (3 + w) in the left-hand side 
and the functions /(z + o>') in the right-hand side. The result 
must be an identity, for otherwise there would be a linear relation 
between the members of the fundamental system f,(2;), ...,/„(«); 
hence, comparing the coefficients of fg (z) on the two sides after 
substitution, we have 



aay. This holds for the m" equations that arise from the values 
r, s= 1, ...,m. Of the m' equations, only m^ — m are independent 
of one another, a statement the verification of which (alike in 
genera!, and for the special values m = 2, m = 3) is left to the 
reader : it can also he inferred from some equations which will 
be obtained immediately. The number of the relations is less 
important, than their existence and their form, for the establish- 
ment of Picard's theorem relating to integrals with the character- 
istic property of doubly-periodic functions of the second kind. 

Consider a linear combination of the members of the funda- 
mental system in the form 

F(^)-\A(z) + KMz) + ...+-K,.f^{^), 

where X, will be taken as equal to unity when it is not bound to 
be zero; and let the constants X^, ..., X^ be chosen so that, if 
possible, the relation 

is satisfied, 6 being some constant. To this end, we must have 
\,0 = \a^^ -l-Xjaai +X3asi + ... 4- X,„a,„| , 



X^0^ 



n + X3%,„ - 



fX,„am„„ 



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141 

and therefore 



FUNDAMENTAL EQUATIONS 



[141. 



the equation satisfied by 0. 

As in the case of the single period in § 128, it may be proved 
that this equation is independent of the original choice of the 
fundamental system of integrals /i(z), ..., fmi^)- The coefficients 
of the various powers of 6 are therefore invariants, and the equa- 
tion is called the fundamental equation for the period o). 

Now let 



= \^1W 



-\A^ + \shsm 



I- \»b,„rn- 



Multiply the earlier equations, which define the quantities X and 
lead to the equation fL(B) = 0, by b^r, b^, ■■■, &«»■ respectively, and 
add: then 

Ofir = \ (Oii fcir + «12 tsr + ■ ■ ■ + tim f>mr) 
+ 'K (<*!! 6,T + «32 6sr + - ■ ■ + Ossn hmr) 



+ Ki (Clnil &ir + ChaAr + ■■■ + amm^mr) 
= \i (b„ a,^ + bi3 a^ + --.+hm O-mf) 



+ Ki (6r«l«]r + h^a^y + . . . + imma,nr) 

This holds for all values of r ; and thus we have 

-nr" = '^ifim + ^ «»ni + ^ ttsni + •■■ + "3'" '^mitf 



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14.1.] 

When the 
uniquely 



FOR THE PERIODS 445 

; compared with the earlier equations, we have 



for all values of r; and therefore the same values of X^, ..,, X„, 
that enable the equations connected with the period fu to be 
:ad to the equations 



X„^=Xi6im + X2ijm + ?^-3&3m+-.-+X^&mm. 

Hence 

J?(2 + «)-X,/,(« + »') + X.,/.(^+«')+--- + V/,.(^ + «') 

- 9' (A,/, w + x,/, (^) + . . . + \,u Wl 



on using the prece 
satisfies the equati 

Q.'{e') = 


ing equations. Moreover, th 
n 

b,^-e\ b^ , ..., b,„, 

b,i , h^-ff, ..., i,„a 


IS multiplier 6 
= 0. 




h«. , b^ i™„-^' 





This equation, like il {B) — 0, is independent of the initial choice 
of the fundamental system of integrals /i (z), .,., /m{^), the proof 
teing similar to that in | 128. The coefficients of the various 
pow&rs of 6' are therefore invariants; and the equation is called 
the fmidamental equation for the period «'. 

The term independent of ^ in II {$), and the term independent 
of 6' in il'{ff), can be obtained simply. Let A (a) denote the 
determinant 

d-'A ■< "/■ d'^A 

da™-' ' ds™-^ dn^-' 



AK 






df, 

dz 

f. 



dU 
dn 



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446 FUNDAMENTAL EQUATIONS [141. 

then, as in § 9, we have 

-.-. — - . 

Hence 

I p, Ix) dx 

so that 

A (z + ») _ f'*"r, (•)<!• 
aw "~" ' 
and similarly 



where we manifestly may assume that the path of integration 
does not approach infinitesimally near the singularities of jj,. 
Now 'pi is a uniform doubly- periodic function with no essentia! 
singularity in the finite part of the plane; if, therefore, a^, ,.,, a„ 
denote its irreducible poles, and if f (s) denote the usual Weier- 
strassian function in the same periods w and a' as Pi, we have* 



with the condition 
Now 

j'*'p, W d^=C^+SA, log ^-(t " v""- 

+ l^ii,{ir(2 + „-«,)-f(^-«,)l 

-Co.+ I ^,|i7r + , ,0 + 2, («-»,)!+ I 2,i(, 
= Co) - 2, S 4,0, + 2, i a, - D, 
' r. 7''., g 129. 



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141,] OF THE PERIODS 447 

say ; and similarly 

say. But, substituting in A (s + w) the expressions for f^ (z + a), 
..., /m (2 + w) and their derivatives in terms of y, (jr), . , . , f^^ (z) 
and their derivatives, we have 









which is the non-vanishing constant term in il (0). Thus 



and similarly 



il'((9') = e»' + 



K-ir^'™ 



In particular, when p-^ is zero, so that the differential equation 
has no term in -^r~ — r , we have I> = 0, D' — Q; and thon 

il(6) = \ + ...+ (- 1)"-^, il'(^')= 1 + ...+(- 1)-"!?"". 

Integrals which are Doitbly-periodic Functions of the 
Second Kind. 

142. Let ^ be a root of the equation SI {&) = 0. Then quanti- 
ties X,, ...,Xm exist such that the equations leading to ii(^) = 
are satisfied; and a quantity 6' is obtained, when the values of 
Xu, ..., X,n are substituted in its expression. It thus follows 
that there is an integral F{z) of the differential equation such 
that 

F{z + a,)^6F{z), F{z + o>') = e'F(z), 

where $ and d' are constants. Such a function is called* doubly- 
periodic of the second kind : and therefore it follows that a linear 
differential equation, which has uniform doubly-petiodio functions 
for its coefficients, possesses an integral which is a doubly -periodic 
function of the second kind: a result first given by Picard. 
* T. F., g lae. 



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448 picard's [142. 

When d is si simple root of the equation fi (S) = 0, then 
Xu, .-.,X™ are uniquely determinate: and 6' is uniquely determ- 
inate. When S is a multiple root of its equation, quantities 
"Kt, ..., Xm exist satisfying the associated equations but they 
are not uniquely determinate : and assigned values of X^, -.-, X™ 
determine ff. Similarly for ^ as a root of the equation li' (6") — 0. 
Combining these results, we have the theorem* i 

A linear differential equation, having doubly-periodic functions 
for its coefficients, possesses at least as many integrals which are 
doubly-periodic functions of the second kind as either of the equa- 
tions li (0) = 0, H' (6') = has distinct roots. 

By using the elementaiy divisors of fl {$) = 0, we can obtain 
a more exact estimate of the number of integrals which are 
periodic functions of the second kind, associated with a multiple 
root. 

Let 6, be a root of il{6) = of multiplicity Xi, and let ii^ be 
the number of different elementary divisors of H (6) which are 
powers of ^ — ^j, so that the minors of il (0) of order m, are the 
first in successively increasing order which do not vanish simul- 
taneously when = $,. Then (§ 133) the number of integrals, 
which satisfy the equation 



IS 



precisely equal to «i. 



* These equfttions appear to have been oonsidered first by Pioard in eenoral; 
see Comptes Remlris, t. sc (1880), pp. 12&-131, 2<)3— 295; Creile, t. xc (1880), 
pp. 281—303. 

Their properties were farther developed by Floquet, Gomptes Bendvs, t. xcviii 
(1634), pp. 82—85, A-rm. de V^b. Norm. Sup., 3™ Sir., t. I (1884), pp. 181—238, 
\Thich should be consulted iu oouueotioii with many of the following investigations. 

A proof of Picard's tbeorem, different from that in the test, is given by Barnes, 
Messenger of Malltematici, t. sxvii (1897), pp. 16, 17. 

Investigations of a difierent kind, leading to equations the primitives of which 
are espreasible in terms of doubly-periodic functions, are curried out in Halphen'a 
memoir "Sur la reduction des equations <liff4rentiel!es lin^aires aux formes 
int^grables," Mim. des Sav. Strang., t. xsvni (1883), No. 1, 301 pp. ; particularly, 
chapters ii and ii. 

The most important equation of the type under consideration is the general 
form of Lainfi'a equation. It had heen considered by Hermite, previous to Picard's 
investigations; and it has formed the subject of many inemoirH, references to some 
of which will be found in my Theory of Fumtiois, ^% 1S7— 141. 



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14;2.] THEOHEW 449 

Moreover, in that case, tii of the equations in § 141 for 
determining the quantities X are dependent upon the remaining 
m — Ui. Let the last m — Ui be a set of independent equations, 
determining Xn^+,, ..., \m in terms of \, Xg, ..., X„^ ; and suppose 
that the expressions are 

Xj = /;,iXi + k^\! + kgs\+ ... + km,'K„ 
for s = «, + !, n, + % ..., m. Then 

F (3) = \A (2) + X./. (2) + , . . + X„./,„ (s) 
= \(fi is) + ^^9^ (^) + . . . + X„, 3,,, {2), 
where 

<,,(»)=/,w+_J^^t/.(.), 

for r = l, 2, ..., n,; and each of the functions g^, ...,^,i, is snch 
that 

j,(2+«.)-e).s,(.). 

Ah regards the possible multiplier 6-^ for the other period, we 
have 

6^ = Xii„ + X2&„i + X,6s, + . . . + X,„J™i 

= x,A + M, + ...+x„,s„,, 

say, where 

and the effect upon F {s) of the increase of argument by the 
period <o' is given by 

Now 6i is not zero, for it is a root of O' {&) = which has no 

zero root ; and therefore not all the quantities B^, B^, ..., Bn, can 

vanish. Let Bi, B^, ..., B, be those which do not vanish ; then we 

have 

X.sf, (s + 0.) + X^g^ (^ + w) + . . . + X„,^„, {z + <o) 

= (XiS, -H X,B, + ... + X^B,) [\g^ {e) + \^g^ (s) + ... + X„,^„, (s)}. 
As some one of the quantities Xj, X^, ..., X^, is not zero (for, thus 
far, all these quantities are arbitrary), we shall take X,= l. In 
order that this equation may hold, we assign definite values to 
Xj, ..., A,; we write 

B, + \B,+ ... + X,B, = 0,', 
g, (^) + X.i'. (2) + . . . + X,5-, (z) = G (^), 

F. IV. 29 



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450 DOUBLY-PEllIODIO INTEGRALS [142. 

and then, as Xg+i, ..., >.«, t^n remain arbitrary, wo have 

forr = s + l, ...,K,. Moreover, on account of the composition of 
Q (z), we have 

and we had 






Accordingly, the nmnber of integrals, which are doubly -penodic 
functions of the second kind and are associated with the multiple 
root 0, of the fundamental equation ii{^) = 0, is 



where % is the mimber vf elemental y divisors of ii(^) which are 
powers of — 0,, "nd s n the number oj quantities 

which do not vanish, so that < s < Ki- 

143. We now can indicate the total number of integrals, 
which are doubly-periodic functions of the second kind. 

Let 6, be a root of multiplicity X, of fi {6) = 0, and let it give 
rise to n^ elementary divisors of fl (8) which are powers of ^ — ^j ; 
and let s, be the number of quantities 

in the preceding investigation which do not vanish, so that 

<5i <«] €X,. 
Let 0^, 03, ... be other multiple roots; and let X^, n^, s^; \^, jt,, 
Sj ; . . . be the numbers for them, corresponding to X, , n, , s^ for 0^ ; so 
that 

X, + \ + \+... = m. 

Then the number of integi'als, which are doubly-periodic functions 
of the second kind, is 

2 (1+M,,_s,.). 



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143.] OF THE SECOND KIND 451 

In particular, if the roots of Cl(0) = O be all distinct from one 
another, a fundamental system can be composed of m integrals, 
each of which is a doubly-periodic function of the second kind; 
the constant multipliers are the m roots of il (0) = 0, and the 
corresponding quantities 0' derived from them, these quantities 0' 
themselves satisfying the equation O.' (ff) = (I. 

Moreover, the relation between the equations satisfied by 
and Xi, ..., X™, and the equations satisfied by 0' and X,, ..., X™, 
is reciprocal ; for each set can be constructed from the other as in 
1 141. Hence, if either of the equations £1 (f) = and li' (0') = 
has all its roots distinct from one another, there is no necessity to 
take account of possible multiplicity of the roots of the other, so 
far as the present purpose is concerned : the implication merely 
is that one of the two multipliers has the same value for several 
of the integrals. 

Further, if 9 and 0' are two associated multipliers, each of 
them arising as repeated roots of their respective equations, we 
shall suppose, for the same reason as in the preceding case, that 
the construction of the doubly-periodic functions of the second 
kind is initially associated with that one of the two equations 
which has the repeated root in the smaller multiplicity. 



Multiple Roots of the Fundamental Equations and 
Associated Interralh, 

144. Wo have now to consider the form of the integrals 
associated with a multiple root of II (8) = 0, the fundamental 
equation for the period w ; and we assume that the correspond- 
ing root of li'(^') = is also multiple, to at least as great an 
order of multiplicity. Denoting this root by 0, and the corre- 
sponding root of il' (6') = by 0', we know that there certainly 
is one integral, which is doubly-periodic of the second kind and 
has multipliers and 0': let it be denoted by .^, so that 

<t>, {z + m) = 04,, (2), 0, {2 ^ m') - e'<i., {z). 

Considering the integrals first in relation to the period m, we 
know (§ 134) that the number of them associated with the 
multiple root is equal to the order of multiplicity of 8: and 

39—3 



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452 INTEGBAL8 ASSOCIATED [144. 

further, that this group of integrals is linearly equivalent to 
sub-groups of integrals of the form 



Wj = 01 + 330, + 3£"0a + S^(j>„ 

the aggregate number of integrals in the various sub-groups is 
equal to the order of the multiplicity of 0, and each of the 
functions is such that 

^(^ + »)-«+ (4 

In these integrals, 0^ can have any added constant multiple of 
01 ; also 05 can have any linear combination of constant multiples 
of 0s and 01 ; and so on. All the functions 0, so changed, still 
have the multiplier 8 for the period w. 

Now u, has the multiplier ff for the period m'. The simplest 
case arises when some other integral of the group, say %, also has 
this multiplier $' for the period a>' : for then all the intervening 
integrals have this multiplier for the period co'. What is neces- 
sary to secure this result is that, first, 

,(., (» + »■) +(z + «,■) *, (, + «■) = »'!*, (^) + ^^, wi, 

that is, 

and therefore 

■ ».(^ + »') ,., ».w 

*,(»+»■) + " *«■ 

Secondly, we must have 

.^, (2 + »') + 2 (« + «■) ^, (« + •>') + (^ + »•)• * (» + »') 

-9'l*.(.) + 2#,W + ^-*(A, 

which, in connection with the preceding equations, is satisfied if 

05 (s + 0.') -i- 2«'0., (s + a.') -H m'^0, (2 -I- «.') = 0-<l>, (z), 

that is, if 



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144.] 




WITH MULTIPLE 


; ROC 


Similai-ly, we 


must have 






4>,(z + (, 


?!-. 


,*.(» + ■ 


'-} + Sa" 


,*.o 


M^ + u 


*.o 


and so on. 











•■) *W 



Let ^(s) denote the usual "Weieratrassian ^-function, with 
periods oi and w'; and let 17, V denote the increments of ^(z) for 
an increase of s hy the respective periods, so that we have 

1701' — Jj'fii = + Stt*, 

the sign being the same as that of the real part of a>' -?- ita. Then, 
if a function u{s) be defined by the equation 



.(»+« 


) = »W, 


«(^ + « 


■)-«« + „'. 




.(.+„■)-*■« 

^.(2) 



"W. 

that is, the function on the right-hand side is periodic in m'. 
Moreover, <J3^ and 0, have the same multiplier for q>, and u {z) is 
periodic in w; hence the function on the right-hand side is 
periodic in w also. It thus is a doubly-periodic function of the 
first kind ; denoting it by i/^.^, we have 



so that ^)i^2 is a doubly-periodic function of the second kind, 
with the same multipliers as 1^1, viz. 6 and &. 



Similariy, we have 

3 that the function on the right-hand side manifestly is periodic 
1 m' ; and it is periodic in co on account of the properties of m and 



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454 INTEGRAI£ ASSOCIATED [144, 

the functions i^. Denoting this doubly- periodic function of the 
first kind by i/fj, we have 

And so on in succession. The group of integrals, in the case 
id, can be represented in the form 



where the functions ^i, ^^, ^s, Xi' ■■■ ^^^ doubly -periodic functions 
of the second kind with the multipliers 8 and 8', and 

145. Returning now to the less simple case, when not more 
than one of the integrals associated with the corresponding 
multiple roots can be assumed to be doubly- periodic of the second 
kind, wc know that one integral certainly exists in the form of a 
doubly -periodic function of the seoond kind with the multipliers 6 
and 8". Denoting it by </ii(s), we use it to replace some one of 
the integrals, say fi(s:), in the fundamental system, which then 
becomes 

*(A /.(A .... /„('}■ 

We have 

*(^ + »)=»*,(<:), 



The fundamental equation for the period to ia 

n(x)^\ e-x, , , ..., =0; 



and so f is a root of 



of multiplicity less by one than its multiplicity for li {x) = 0. 



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145.] WITH MULTIPLE ROOTS 

Similarly, we have relations of the form 

/, (3 +<c') = d^ 4,, (z) + d.^f.A^) + --+d^ /™ (^X 

the multiplier ff being a root of 

D.{{x) = -d.^-x, d^, ..., d,„, 1 = 0, 



of multiplicity less by one than its multiplicity for Xi' (x) = 0. 

The coefficients cj and dij must satisfy conditions in order to 
have 

/.|{. + «') + <.|=/.l(^ + <.) + «'!. 

for all values of k : these conditions are 

dn(hi + 'ir3Cig + ... + drmCrm=^ Crldu + Cndis + ■■■ + Crmdms 
= K. 

say, with the limitations 

Cn = 0, d„ = d'; c.s = 0, d,, = 0, it s>l. 
Owing to the fact that ^ is a root of Oi(,k) = 0, quantities 
*3, ..,, Km exist such that 

^Ks^Cas + CssKa-l- ... +C,„sK„,, (s = 8, ..., m), 

^ = Cai + Cja/ira + ... +Cm2«m- 

Let 

'^ = dss + dsiKi+ ... +d„,^K^, 
iTf-d^+dirK3+ ■■■ + datrK„. {r = 3, ..., m); 
then 

Sa-p = dir (c-a + Csa K3 + ... + Cms «m) 
+ '^3r (Csa +C33 Ka+-.- + Cm3 Km) 



(^2 Car + "^ Csr + . - . + daiH Cmr 
+ Kj (das Cffl. + rfa3 «»+■■• +'^sm Cmr) 
+ 

+ «™ (<4.aCw + C^iaCar + ■ . . + (4™C™.) 
; CjA + Car ffa + CirO-4 + . . ■ + C™.(7™ ; 



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MULTIPLE ROOTS AND [145. 



holding for all values of r. Comparing with the oarlier equations 
in c, we have 

0-, - ?»/<:,., 

for all values of r ; and thus the second set of equations is 

a- = rf^ + (;^Ks + ... + <^ffl3«m, 

^/(:,.= (4. + 4,«:a+ ... +d,„K„, (r = 3, ...,m). 

Eliminating the quantities rc, we have 

so that 0' is the value of a. 
Now consider the integral 

X. W -/. w + «./, (2) + . . . + «./.. (t). 

We have 

X. (» + »)-/■ (2 +«) + «./,(» + ») + .•.+«,./,.<« + ») 

= ".*« + "»(«), 
say, and 

» (' + »■) -/■ {•:+•>') + «./. (« + »•)+... + «./„ (^ + »•) 

= 6,*,W + «'X,«, 
say. When Hj and 62 vanish, ^^ is doubly-periodic of the second 
kind; but in the general case, a.^ and b^ are distinct from zero. 
The property 

x,l(^ + ») + .1-»|(^ + «') + »! 

leads to no relation between 0,3 and b^. 

If the multiplicity of (9 as a root of 12 (^) = and that of 6' as 
a root of H' (^) = be greater than 2, so that and ^' are 
multiple roots of ilj («) = 0, flj (x) = 0, we proceed as above. The 
newly obtained integral j;, is used to modify the fundamental 
system by replacing y^, say, so that the system consists of 

A. X.. /. /»■ 



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145.] ASSOCIATED INTEGRALS 457 

Then, in the same way as above, it is proved that an integral ;:^ 
exists such that 

Since 

»K^ + .) + »■) -»!(« + »') + »), 

we find, on aubstitation, 

so that we may take 

c^ — Xa^, di^Xb^, 

where \ is any parameter. This parameter may clearly be 
absorbed into j^, by taking Xa-^V and also into a, and h^ by 
division. Thus our integrals 0i, ')(_^, j^j are such that 

Xa (s + » ) = 050, + a^x-' + ^X3. 
<^,(^ + a,') = '5'0„ 

And so on, until a number of integrals is obtained equal to the 
leaser of the orders of multiplicity of 6 and 6'. Thus the next 
integral is x* (say), where 

X. (^ + « ) = a,0i + («-j + ^o.) X= + a=Xs + ^X- . 
X, (s + <o') = b,<f>, + {h + \h) X, + b,x, + B'x. . 

146. From these descriptive forms, we can proceed one stage 
towards the construction of an analytical form of the integrals. 
For this purpose, we introduce (as in § 144) the functions 

where the doubtful sign is the same as that of the real part of 
w' -^ i<it ; we have 

Then the various integrals can be expressed as non-homogeneous 
polynomials in i( (s) and v (z), the coefficients of which are doubly- 



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458 ANALYTICAL FORM OF INTEGRALS [146. 

periodic functions of the second kind, with and 0' for multipliers. 
In particular, the integrals have the form 

X, » = *-,(.) + /.*■,(«). 

x,W-*'.W + -r.-*'.W+W-f.« + K.*',(4 

and so on. The functions F are doubly- periodic of the second kind 
with factors 8 and &; /, is a polynomial of the first degree in u (s) 
and v(z); /a and J^ are polynomials of the second degree in the 
same quantities, having I^ as the aggregate of their terms of the 
second degree ; /j is a polynomial of the third degree in the same 
quantities, having I^ as the aggregate of its terms of the third 
degree; and so on. 

To prove this, we note in the first place that </>i {z} is a doubiy- 
periodic function of the second kind with the multipliers 6 and ff. 
As for Xi (■2)1 ■'^e have 

?6(g + <^) _ Xi(£) , ^ 











If therefore we take the function p,, (z), where 






?.,(-). J. 


W + j1«W, 




we have 

*(» + . 


iy-M'*" 




W 


and therefore 




^ »(» + «') 
-ft.(2> 


-P21O 



is a doubly -periodic function of the first kind. Let -^2(2) denote 
the product of this function and 0i (z) ; then F^ (z) is doubly- 
periodic of the second kind with multipliers and 8', and we 
have 



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146.] ASSOCIATED WITH MUtTIPLE EOOT 459 

Similarly, if 

...«^{i(ty--}.w.|i(|)"-^}«(4 

we find 

to be a doubly- periodic function of the first kind. Let F^{z) 
denote the product of this function and 0i(3); then Fi(z) is 
doubly-periodic of the second kind with multipliers and &, and 
wc have 

X. (^) - ^. W + p. (') f. W + iiP=." (^) - r- (')! * W- 

Similarly, after reduction, 

X, (.) = -F. W + p„ » ^, («) + (ip.- W - ft, (2)} -f. W 
+ (iP.'W-p..W!*.W 

where -Fj (z) is a doubly-periodic function of the second kind with 
multipliers 6 and $', pt, (s) is a polynomial in u and v of the first 
degree, and p^ (2) is a polynomial (not homogeneous) in u and v 
of the second degree. 

And so on, in general : the theorem is thus established. 

Construction of Integrals that are Uneeokm. 

147. Further progress in the efifective determination of the 
analytical forms of the integrals on the basis of the foregoing 
properties is not possible in the general case. When particular 
classes of limitations are imposed upon the coefficients in the 
original differential equation, such progress might be possible : 
but it frequently happens that some more special method loads 
more directly to the solution. 

The simplest case is that in which the equation possesses a 
uniform integral, or in which the equation has several uniform 
integrals: but, of course, the preceding investigations in §§ 141 — 
146 apply to all equations of the type considered, whether they 
have uniform integrals or not. When all the integrals are 
uniform (and this can be determined independently by consider- 
ing their forms in the vicinity of the singularities), then the 



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460 EQUATIONS HAVING [147. 

doubly -periodic functions of the second kind arising in the pre- 
ceding investigation are uniform functions of z; and a general 
method of constructing such functions is known*. Instead, how- 
ever, of using the preceding results, it sometimes is more con- 
venient and more direct to infer the irreducible singularities of 
the integrals from the differential equation itself. These are used 
to construct an appropriate uniform doubly-periodic function of 
the second kind; the remaining quantities needed for the precise 
determination of the integral are then inferred by substituting 
the expression in the differential equation. 

Ea:. 1. Consider the equation t 

with the usual notation for the Weierstrassian elliptic functions; q and ^ are 
Constanta. 

The only irreducible singularity that an integral can have is 2=0. The 
iodicial equation for e=0 is 

m(»-I)(«-2)-6«=0, 
the roots of which are -1, 0, 4; and the expansions that respectively corre- 
spond to the roots are easily proved to be 

Wj-l-l- 1^(33^-1-^503!^ + ..., 

Thus no It^arithniB are involved ; every integral is a uniform function of z, 
being of the form. Aw^-\-Bw^-\-Ow^; and at least one integral of the equation 
is thus a uniform doubly*periodic function of the second kind. We proceed 
to its construction. 

This doubly-periodio function of the second kind cannot be devoid of poles, 
if it is to involve the first of the above integrals in its expreasion. (If it were 
devoid of poles, it would alsoj be devoid of zeros in the finite part of the 
plane : and then {I.e.') it could only be an exponential of the form e", which 
is manifestly not a solution of oiu' equation.) It has one irreducible pole ; it 
therefore has one irreducible zero in the finite part of the plane. Let the 
latter be denoted by - a, which at present is unknown. 

We now consider § the elementary function 
" (■' + «) M 

* T. J., g§ 137—139. 

+ It ia a modified form of an eiiuation given by Picai'd, C'i'dle, t. kc, p. 290. 

t T. F., i 139. g T. F., U. 



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147.] UNIFOKM PERIODIC INTEGRALS 461 

which has £=0 for an irreducible simple pole, and -a for an iiTeducible aiinple 
Kero; its oxpansion begins with s~i, and the function muat therefore agree 
with the integral above obtained in the vicinity of 3 = 0. (The constants X 
and a determine, or are determined by, the multipliers of the periodic function ; 
but at present these are unknown, and so X and a must be detormined in 
another maimer.) To expand the above function in powers of 3, we have 

the Weieratrassian functions on the right-hand aide being functions of a. 



»(.)- 






-J+(>.+0+i'{(»+0'-B+i''»+0"-3(i+f)»'-rl 

+£»+o*-8(^+f)'f-"ip+nf -*■+•«)+•■■ 

This is to satisfy ttie dilferential equation, so tbat it must be of tlie form 

Clearly J = 1, iJ = X + f, for this purpose: the value of P would be needed for 
the complete expression, but we merely require X and « at present. Com- 
paring the coefficieutB, we thus have 

^ = 1, 

a-x+C, 
^— (x+O'-f, 

so that X and a are determined by the equations 

(X+O'-J'— 1 

(x+o'-3(x+f)t>-p'.jfir 



-•V 



x+f. _ 

where a is determined bj' the relation 

Sjy + (3a>-j,)J> + J|3'-a'-J,.0. 

The function on the left-hand side is a doubly-periodic function of tho first 
kind: it has a single irreducible pole, which is at 3=0 and is of multiplicity 
three. Hence it has three irreducible aeros, say a^, a^, a^; and their sum is 
congruent to 0, bo that we may take 

a^ + a^ + a^^O. 



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462 EXAMPLES [147. 

In general, u^, Oj, «j ure unequal, because a and /i are general constants; the 
discussion of the critical conditions, that lead to equalities between a,, a^, a^t 
and of the consequent modifications in the complete primitive, is left as an 



K- S W+ a„2g)(a,) ' ('-1.2,3), 

^^^)^ 

is an integral of the equation for each of the three values of r. Tho primitive 
of the equation is 

whore A-^, A^, A^ are arbitrary constants. 

Jix. 2. Obtain the relations which express the integrals w^, w^, v>^ of the 
equation in the preceding example in terms of H";, W^, W.^; and determine 
the multipliers of the integrals. 

Ew. 3. Obtain the primitive of the equation 

in the form 

w=Ae<^'^ + Bf{z). 

Ex. i. Verify that the primitive of the equation 



da''^ dn. 



y=^cos(nam^)+£sin{»iam:^). 



Ex. 5. Prove that, if / be an odd function and J^ be an even function, 
both doubly -periodic in the same periods, the integrals of the equation 

rfe= ds 
oan bo expressed in tenns of JJ (z). 

Hence (or otherwise) integrate the equation 

cr ' r 

J^x. S. Determine tbe relations among tlie constants (if any) in tlie 
equation 

»'"+(a-SB»'+(y+(iJ)-|j)>.0, 

in order that every integral of tlie equation aliould te uniform ; atid assuming 
tile relations satisfied, shew that the equation has three integrals of the form 

'(■+") ^ 



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147.] EXAMPLES 463 

Ha. 7. Shew that the equation 

has an integral of the form 

^""^ ®(^) ^"^ "' 
provided 

3a + / = 3(l+i2). 

and that it then has three integrals of that form. Obtain these integrals. 

(Mitte^-LefBer.) 
Eu-. 8. Obtain the integral of the equation 

in the form 

^- ■&(:.) 
where the constants X and « ai'o given bj the equations 
A-(14-(t=) + 3(A=-i^8na«) = 0, 
2X=-6Xi"8n^B + 3X(H-F)-4i2sn«)cnQ.dnco-Ai=0. 
Verify that, in general, three distinct integrals are thus obtained. (Picard.) 
A'iB. 9. Prove that the equation 



has an integral of the form 

^ &(ic) ' 

provided 

2a + S = 8(l+^)i 
and that, if this relation he satisfied, it has fonr such int^rals. Obtain 
them. (Mittag-Loffler.) 

Em. 10. Verify that the equation 
(m-.-.,l'<.)g-2.n>:«oxdn^J + 2j|l-2(I+i').n-,. + 3f .■,•.).! 
hiia an integral of the form 

provided 



an^'ci-sn^B ' afc2sn*K-2(I+i=)sn«a + l ' 

and obtain the primitive. 

Hence integrate the equation 

where A is a constant 



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464 lamp's [147. 

Ew. 11. Discuss the equation 

S + |-2J,(.))*„J + ,, + ,p(.,,..„, 

for those cases when every integral is a uniform function of z. 

Ea:. 12. Shew that there are three sets of values of the constants o, and 6, 
for which the equation 

33-(Ar"(»)+ir (»)+"?(")+»)? 

admits aa an integral a uniform douhly -periodic function ; and obtain the 
integral (Math. Tripos, Part ii, 1997.) 

Ex. 13. Prove that the equation 
y'"-2»(K + l)y'p(2)-2w(ft + l)/^'(.) 

where a ia an arbitrary constant and n is a positive integer, has a uniform 
function of z for its complete primitive. (H^lphen.) 

Ex. 14. Construct the equation which has 

for its complete primitive and, for a properly determined value of /(a), is 
devoid of the term in -^ . Likewise construct the equation which has 

w={a,+agsn3 + 0!3cnj+aidns}/(j) 

for its complete primitive, with the corresponding determination ai f{z) to 

remove the term in -^ . In each case, the quantities a^, a^, %, a^ are to be 

regarded as arbitrary constants. (Halphen.) 

Ex. 15. Prove that the primitive of the equation 

is a uniform function of z, when m is an integer multiple of 3 ; and discuss 
the primitive, when the integer n is prime to 3. (Halphen.) 

Lame's Equation. 

148. One of the most important instances, in which a dif- 
ferential equation with uniform douhly-periodic coefGcients has 
a uniform doubly -periodic function of the second kind for its 
integral, ia Lamp's equation or, rather, the more general form of 



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148.] EQUATION 465 

Lamp's equation as discussed in the investigations of Hermite, 
Halphen, and others. The form used by Hermite* is 



where n is a positive integer and .S is a general constant; the 
form used hy Halphenf is 

with the same significance for n and B. We shall use the latter 
form of equation ; it is selected for convenience and for its slightly 
greater generality owing to the functional independence of the 
The mode of discussion is the same for the two forms. 



As we are concerned with the application of the general 
theory!, rather than with the special properties of the functions 
defined by Lamp's equation, only an outline of the solution of the 
equation will be given here. The detailed developments, and 
references to further memoirs, will be found in the authorities 
just quoted. 

It may bo not without interest to indicate how this form of 
equation arises from the equation 

da? dy' d^ ' 
characteristic of the potential in free space. When orthogonal 
curvilinear coordinates a, /3, 7, as defined by three orthogonal 



are used, then the equation becomes 

do:[BC Sa>'^Sg\CAafil St\AB dy) • 

* " Siir que <i e apjl at on deE foBctions elliptiques, " a separate repiitit 
(1B85) Irom the C np He E s 

+ Traite dts to ctio e Ipt q es t ch. xir. 

J That is the theory of the un f rm doubly -periodic functions of the second 
kind which are ntegials of tlie differ nt al equation. It has been proved (g 54) 
that, by an app 01 t anal in on the equation can be changed eo as to be of 

Fuehsian type 

I. IV. 30 



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466 SOURCE OF [148. 

where 

«-©■-©■-©■• 

Choose, as the orthogonal surfaces, the three quadrics which are 
coiifocal with a given ellipsoid ; and let \, fi, v be the roots of the 
equation 

ir} if ^° _ -I 

a cubic in 9. Then* we take 

([2 + \=:^(a) - ^1, ¥ ■\-\^f{oL) - e.,, c^ + \ =g>(a) -- e^, 
«' + ^ = ^(^)-e„ 6^ + M = F(/3)~e.„ c-^ + ^ = g)(^)-e„ 

Now 

jd\\^ /dXV /3XV 



Kdcc) ^ \dyl ^ \dz} 



- ^'^: 



where 



\ _ g? y s^ 

jiT" ^ (a' + Xf + (6" + X)" "'" (C + X)' 

(X-y)( X -») 

((•■ + X)(6" + X)(c" + X) 

!>■■(«) 



1 „, 1 



Similarly 

so that the equation for the potential becomes 

* Greenhill, Pi-oc, ioiid. .Viif/f, Soc, t. xviii (18B7|. p. 27S. 



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148.] lamp's kquation 467 

or, what ifi the same thing, 

IP(«-S>(7)lj„'^+l!'(7)-?WlU+[S>W-p(ffll|^ = 0. 

For the purposes contemplated in the transformation, the quantity 
V is the product of a function of a, a function of /3, and a function 
of 7, or is an aggregate of such products ; and it is a uniform 
function of its variables. Hence, writing 

F=/(»)!,(/3);.(7), 

where /, g, h denote uniform functions of their arguments, we 
have 



where w =f when z = a, w = g when z=^^, w = h when s — y, and 
A, B are constants independent of a, /3, 7 : they must be such as 
will, if possible, make v) a uniform function of its argument. The 
only possible singularities of w are ^ = and points congruent 
with z=0; hence, after the earlier investigations, we consider the 
irreducible point 2=0. The form of the equation shews that it 
will be an infinity of w; and thus it must be a pole, say of 
order n, where n, is, & positive integer. Thus we have, in the 
vicinity of the pole, 

«; = J + -,^^,+ ... =^-K(s), 

where R (2) and iJi {z) are regular functions of z, such that 
BA£)_ 

Hence, in the vicinity ni z — 0, we have 

- -j-^ =— -^ — ^ — ' (1 + powers 01 z), 



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468 INTEGRATION OF [14S. 

and therefore 

^-.(n + l). 
a limitation upon the form of the constant A. But there is no 
limitation upon B, necessary for the existence of integrals of the 
type indicated ; and therefore the differential equation may be 
taken in the form as stated. 

To obtain Hermite'a form, we write 

, , _ «! — 63 _ - ,^ J _e.^ — e^ 

f {s) - 03 - gnSy ' V- ^(^1- ^3>. ■ - g^ _ g^ • 

as usual, and then take 

y = cc + i'K' ; 
the equation becomes 

where B" is a constant. 

14!). The method of solution of the equation is based upon 
the knowledge that there is at least one integral in the form of a 
doubly-periodic function of the second kind ; the limitations, that 
have been imposed upon the equation, secure that this function is 
uniform. Moreover, the integral has only one irreducible pole, 
viz. at s = 0, and the pole is of order n. 

There are two modes of using these results in order to 
construct the integral. 

By one of them, we use* the further property that a uniform 
doubly- periodic function of the second kind has as many irre- 
ducible zeros as it has irreducible poles, account being taken of 
the orders of the points in each category. Accordingly, in the 
present instance, the integral has n irreducible zeros : let them be 
— rii, — «a, .--, — <^ij- Consider the uniform function 

which is doubly- periodic of the second kind; its (single) irre- 
ducible pole is of order n and is at s = 0; and it possesses the 

* r. F., %% 139, 141. 



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149.] lamp's equation 469 

necessary « (unkiiown) irreducible zeros, so that it is of a suitable 
form. We have 

1 __ 

w dz 
In order to simplify the right-hand side, it is conveoient to take 

and so 

Hence 

1 d?w 1 idwy , , o / , 



and therefore 



1— _x+i I y'w^ p'w 



li)(«,)-yW( 
.lii y'(n,)-p'W p'(a, ) -y'W 

Tlie first term on the right-hand side is equal to 

To modify the second term, where the summation is for pairs of 
unequal values of r and s, we have 

li>'('V)-y'W P' (»■)-»'(») 
(>("»)-pW ■ «>(«.)-«'(»> 
_ yW- g.pW-g.-p'(»r)y'(''.)-f'MI»'K) + t''(».)l 



after easy reductions, where 
i,. 



?(«)-!>("-) PW-PK)) 



. P' (»>) + P' (a. ) . 
' P(»r)-S)(».) ' 



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470 lamp's equation [149. 

and thus the second turm in the expression for —^\-j-] becomes 

»{,.-i),,(.)+2(..-i)_^ls.(«.,)+i_|f'<|>f^'l;i,,. 

where the summation is for all values of 5 from 1 to w except only 
s = r. Then 

^ | y(a..)-y-W {f,^ aJ, 

mtW-fM U-i ) 

Comparing this result with the differential equation under con- 
sideration, we naturally take 

i' L,, - 2\ 

for all values of r, that is, 

»'M + f'('h) P'W + f'l". ) ^ _ 2^^ 
j>((i,)-i)(ii,) J)(o,)-|>(o,) 

f'W + y'W , y'(».) + t^(».) , _ 2x 
fW-|>(«.) «>(".)-«><».) 

f'(°~) +»'(■■■) ^. g'W + y'W ^ _ 2^ 
«'(».)-8'(»i) «'("•)-(('(«!) 

and (2" - 1) ^ g* («.) + X^ = 5. 

Adding the n former equations together, we have 
= 2n\, 

so that \ vanishes. Hence if the n quantities di, a^ a„ o.re 

determined by the equations 

y'W + y'W y'W + »''('■.) ^ _ „ 

(f'W + f'W , y'(».) + y'(a.) ^ ^ D 
?(".)-(»(»■) fW-pW 

{which are equivalent to only k — 1 independent equations, becaii^e 
the sum of the n [eft-hand sides is zero) and by 



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119.] 




XNTEGKATED 


thm 








F{z). 


a{z + a^)„{z + a.i...<, 




»-w 



is an integral of Lamp's equation 

i^ = «(« + l)^(2) + B. 

w dz^ ' 

The equation remains unchanged when —2 is written for z; 
hence F{—z) is also an integral. Save in the case when the 
constants a are such that F{z') and F{-z) are effectively the 
same function, we have two independent integrals of the equa- 
tion, which therefore is completely solved. 

150. Another method of arranging the necessary analysis is 
as follows. Consider the equation 

where F(z) is a doubly- periodic function; by Picard's theorem 
(§ 142), an integral (say Wi) is known to exist in the form of a 
doubly-periodic function of the second kind. If then we write 

_ 1 dwi 

the quantity w is a doubly-periodic function of the iirst kind ; and 
it satisfies the equation 

The irreducible poles of w,, in their proper order, are known from 
the singularities of the original equation ; let them be jt in 
number, account being taken of multiplicity. Then each of them 
is a pole of v, of the first order ; and the sum of their residues for 
vis —n. The number* of irreducible zeros of w, is also n, account 
being taken of multiplicity; each of them is a pole of v, of the 
first order, and the sum of their resid\ies for v is +n. 

We therefore construct a uniform doubly -periodic function of 
the first kind, having these poles, all simple, viz. the known poles 
arising through the singularities of F, and the unknown poles 



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472 EQUATIONS HAVING [150 

arising through the zeros of Wi, taking care to have —n and +n 
for the respective sums of the residues. The general expression 
for such a function is known*: when substituted as a trial 
function in the above equation, comparison of the results leads to 
a determination of the constants. 

As an illustration, consider the equation 
1 iPw „ , , T. 

The irreducible pole of g) (e), viz., 2 = 0, is the only irreducible 
pole of Wi, and it is of the first degree. Accordingly, it is a 
simple pole of v, with a residue —1. Further, there is (by the 
preceding argument) only one other pole of ii : it is simple, and 
has a residue + 1, As w is a doubly- periodic function of the first 
kind, an appropriate expression is 

.i:(«-c)-fw + f(c) + i., 

say ; and b, c have to be determined by substituting in the 
equation 

* + .. = 2p(.) + a 

Now 

dv , , / i 

and by the addition- theorem for the f-function, we have 

'' + *j)W-j.(o)- 
Substituting, we have 

if W + f(c) + b- + b ^>-+ ^'l-) = 28, («) + B, 

which must be satisfied identically. Accordingly, 

6 = 0, ^(c) = B; 
and thus, with a known value of c, 

^ = j:(2-c)-r(5) + f(c), 

* T. P., S 138. 



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150.] DOUBLY-PERIODIC COEFFICIENTS 473 

SO that 

There are two values o£ c, equal and opposite : the construction of 
the primitive is immediate. 

Ex. 1. Shew that two independent integrals of tiie equation 

ill the case when 5 = ej, are given by 

|y(.)-,,(', (»>(.)-.,}»{(■(.+.)+.,.)! 

and obtain the integrals in the cases, when B^e^, and B = e^, respectively. 
K:'). 2. Obtain the primitive of the equation 



(where q i.s constant), in the form 

where 

DiacuaM the solution in the three particular cases 

<i = I+F, 1, lc\ (Ilermite, 

Ex, 3. Shew that 

satisfies the eqviation 

if ^ ("i)' P ("2)1 ^ ("3) "^^ *hs roots of the oubic equation 

and deduce the primitive. (Ilalphen. 

Ex. 4. Shew that the primitive of the equation 

can be expressed in finite form for appropriate values of the constant B it 
the following cases : — 

I. When *i is an even integer, =2»i, then either 

where e^^, e are any two of the throe constants e^, ^j, e^ : 



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474 EXAMPLES [1 50. 

11. when )t is an odd integer, =2(1? — 1, then either 

«'={PW-«;.}*{c™_,iy"'-'(^)+...+C„}, 

where e^ is any one of the three constants e„ e^, e^. 

Determine the number of sohitions of the specified kind in each of the 
cases indicated. (Crawford.) 

Ex. 5. Shew that an integral of the equation 

where h is a coJiatant and m, n are integers, can he expressed in the form 



_,»,(»-.,) a, (.-.,).. .8, fr-....) 



z zK) 



in the usual notation of the theta-ftinctions, ^,, z^, ..., ^m-m hcing appropriate 
constants. 

Obtain the primitive. (M. Elliott.) 

Bx. 6. Obtain the primitive of the equation 

»-4Ji%)<*«"<''-''»*W-« 

Ea:, 7. Shew that there are two values of ij, for which the equation 

i5-i.+i,J>»-2~r(') + t""S''». 

where m is a constant, possesses an integral of the form 

^ ,-.-t.;(.|.(«-i'). 
,(■) ■ 

and, for each such value, obtain the primitive. (Bonoit.) 

-fie. 8. Shew that there are m+l values of k^,, for which the equation 

where m is a constant and n a positive integer, possesses an integral of the 

Prove abo that, if the right-hand side of the differential equation be 
increased by * (s), where * is a doubly-periodic function of the first kind 
having all its poles simple, a corresponding theorem holds as regards the 
integral, if it,, be properly determined. (Benoit.) 



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150.] ALTERNATIVE PROCESS 475 

Es. 9. Integrate the eqiiation 

where n, n', n", n ate positive integers. (Darboui.) 

151. The other mode of utilising the known properties of the 
integral, when it is a uniform doubly-periodic function of the 
second kind, is to obtain the actual expansion of the integral 
in the vicinity of its irreducible pole and thence to construct its 
functional expression in terms of the elementary function 

»(,) ■ 

where a and \ are initially unknown constants. Some indication 
of the process is given in Ex. 1, § 147; but a slightly different 
form will be adopted for the present purpose. We take the 
elementary function in the form 

'{' + 

where p and a are now to be regarded as the constants to be 
determined. The expansion of this function in the vicinity of its 
irreducible pole at s = is 

+ sV {p' - 6/>=F («) - ^99' («) - W («) + *?=} ^ + ■ ■ ■ ■ 

If, in the same vicinity, an integral of the diiferential equation 
exists in the form 

. . . + ' + (to + positive powers, 
then we rnay take 

where a comparison of expansions serves to determine the con- 
stants a and p. The integral thus is known. 



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476 EXAMPLE [151, 

An illustration will render the details clearer. In the case 
when 91=2, the equation is 

1 d'^J) „ , , n 

Let 

1 a, 
w = -^ + - + (ti + ctjs + a,s^ + . .. 

be substituted in the equation ; we find 

a,= Q, ((,, = 0, a,-0, ... 

a,- -IB. a, = i-,]P-i^g„... 
so that 

Manifestly, the form to take is 

__dG_ 
dz ' 
and then eompai-ing the two expansions, we have 

-ilp'-p(«)l--i-B. 
/-3(jy(»)-p'(») = 0. 

These equations give 

BM--27y. 



3y'(a) 

The former in general leads to two irreducible values of a ; the 
latter uniquely determines p for each of these values of a. 
Denoting the two values of a by a and — a, and writing 

<r(«)<r(o) 

the primitive of the differential equation is 

... r<ie. . ..dc. 



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151.] EXAMPLES 477 

Bx, 1. Discuss tho integral of the equat.ioii 

when a, as obtained in the preceding solution, has the values 0, <u, m', lu", 
respectively. 

Ex,. 2. Prove that an integral of the equation 
in given b, 

the constants p and a being given by the equations 






Deduce the primitiv' 



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CHAPTER X. 

Equations having Algebraic Coefficients. 

152. The differential equations, considered in the preceding 
chapters, have had uniform functions of the independent variable 
for their coefficients. We now proceed to consider (but only 
briefly) some equations without this limitation : one of the most 
important classes is constituted by those which have algebraic 
functions of the variable as their coefficients. For this purpose, 
let y denote an algebraic function of the independent variable 
X, defined by the equation 

where 1^ is a polynomial in x and y, and the equation is of genus 
p. With this algebraic equation we associate the proper Eiemann 
surfiwe of connectivity 2j) + 1. 

We assume that the linear differential equation has uniform 
functions of x and y for its coefficients, so that each of these is 
a uniform function of position on the surface : and we write the 
equation in the form 

'""" + a, (a^, y) ;^ + a, (^, !/) ^:_~ + . . . + a,„ («^, ;/) ^^ = 0. 



rf^™-^"=^-^-^^rf^' 



„=Jf 



:, y) dx 



the exponential in the factor of u on the right-hand side being an 
Abelian integral ; then the equation for w is 



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152.] ALGEBRAIC COEFFICIENTS 479 

devoid of the derivative of order m — l; all the coefficienta 
Pa, ■--. Pm ^>^^ algebraic functions of x, and are uniform functions 
of X and y. This is the form of equation which will be discussed. 
Let (iCj, jio) denote any position on the surface, which is not a 
singular point on the surface and in the vicinity of which each of 
the coefficients P is regular. Then an integral exists, which is 
regular everywhere over a domain in the surface, and is uniquely 
determined by the assignment of arbitrary values to iv and to its 
first m — 1 derivatives at (ic,,. po)- I" f*ct, all the results relating 
to the synectic integrals of an equation with uniform coefficients 
hold for the present equation in the domain of (x^, y^. 

Next, let account be taken of the singularities of the equation 
■>|r = and of the associated surface. As these affect all the 
coefficients of all differential equations of the class considered, 
and thus afford no relative discrimination among the functions 
defined by those equations, we shall assume them simplified as 
much as possible before proceeding to consider the properties of 
the functions. Accordingly, we shall suppose that, if the equa- 
tion -"It = {or the Riemann surface associated with it) possesses 
a complicated singularity, it is resolved* into its simplest form 
by means of birational transformations, so that we may write 

where ff and h are uniform functions which, in connection with 
^ = 0, admit of uniform expressions for f and 17 in terms of 
«— e and y—f, and are such that ^=0, »j = is an ordinary 
position on the transformed Riemann surface. The positions on 
the surface, that have to be considered in connection with the 
differential equation, are now ordinary positions : and therefore, in 
dealing with the theory of the equation, no generality is lost if we 
assume that the singularities of the equation 1^=0 and of the 
Riemann surface are ordinary positions for the integrals. (Of course, 
in any particular example, it may happen that a multiple point on 
the curve ^ = 0, or a branch-point of the associated surface, is 
definitely a singularity of the equation. In order to discuss the 
nature of the integrals in the vicinity of such a point, we takef 



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480 FUKDAMENTAI, SYSTEM [1,52. 

where ]) and q are integers, and S is a holomorphic function of 
its argument that does not vanish when f=0; and then we 
investigate the character of the integrals in the vicinity of 
f-O.) 

Lastly, let (a, b) denote a position on the Riemann surface 
(being a pair of values given by the differential equation) such 
that the coefficients of the equation are not regular in the 
immediate vicinity of (a, b) ; after the preceding explanations, we 
may assume that ?/ — 6 is a holomorphic function of x — a in the 
immediate vicinity of the position. The character of the integrals 
in that region is determined, after substitution oi y — h in terms 
of x — a, in association with an indicial equation ; and the general 
processes of the theory, in the case of differential equations with 
uniform coefficients, are applicable to the integrals in the vicinity 
of {a, b). As in that earlier theory, we have a fundamental system 
of integrals existing at any ordinary position on the surface, the 
system being composed of m linearly independent members. 
Continuation of these integrals is possible : and by taking all 
admissible paths iirom one ordinary position to any other ordinary 
position (care being taken to avoid the actual singularities), and 
assuming an arbitrary set of initial values at the first point, we 
shall obtain all possible integrals at the second point. Similarly, 
by taking all possible closed paths on the Eiemann surface, 
which begin at an ordinary point (.x^, ya) and return to it, we 
obtain new integrals at the end of the path ; and each of these 
integrals is linearly expressible in terms of the members of the 
initial fundamental system. 



A Fundamental System op Integrals, and the 
Fundamental Equation. 

153. Let Wi, Wi, ,.., Wm denote a fundamental system at 
an ordinary position («o, y„); and let the variable of position 
describe a closed path on the surface returning to (a:„, 7/0), this 
closed path being chosen so as to include the singularity (a, h) 
but no other singularity of the differential equation. Suppose 
that the effect upon the fundamental system, caused by this 
variation of the variable of position, is to change it into 



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153.] 



OF ISTEGRALS 



481 



w^, w^, ..., Wm : then, as in the case of uniform coefficients, the 
latter set also constitute a fundamental system, and the two 
systems are related by the equations 



Mj/= 2 a^^w^, 



(m = i, 



ni\ 



where the determinant of the coefficients a is different from zero. 
This determinant is (as in § 14) equal to unity. For let A denote 
the determinant of the fundamental system 




and let A' denote the same determinant in relation to the 
fundamental system w' ; then, if A denote the determinant of 
the coefficients a^^, we have 

A' = ^A. 



Now, because the term involving the {m 
is absent from the differential equation, wi 



— l)th derivative of w 
have, as in § 14, 



where C is a constant. Let the function A, which is equal to C 
in the vicinity of (a:^, y^, be traced along the closed path which 
the variable of position describes on its return to {xa, y^\ it is 
steadily constant, and its final value is A', so that 



^ = 1. 
Further, as in the cases when the coefficients are uniform 
functions of the independent variable, it is possible to choose a 
linear combination v of the members of the fundamental system 
such that, if if denote the value of v obtained by making the 
variable of position describe the afoi'esaid closed path, we have 



31 



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482 FUKDAMENTAI. KQUATION [153. 

The multiplier ^ is a root of the equation 

«2i , Hsa — ^ flam 



= 1 + I,B + 1,0' + ... + i^-,^-' + (- l)-"^"" = 0. 

This equation is independent of the choice of the fundamental 
system, so that its coefficients may be regarded as invariants of 
the linear substitution, which the fundamental system undergoes 
in the description of the closed path round («, b). 

154. If some, or if all, of the integrals in the vicinity of (a, b) 
are regular in the sense of § 29, then an indicial equation for the 
singularity exists; and if p be a root of this equation for an 
integral with a multiplier 8, then 



If no one of the integrals is regular, there is no valid indicial 
equation. In the first case, the general character of an integral is 
determined by the value of p : and the explicit form is obtained 
by substituting an expression of the appropriate chai'acter so as to 
determine the coefficients. In the second case, various methods* 
for obtaining the value of S have been suggested, by Fuchsf, 
Hamburgerj, and Poincare§; the most general is the method of 
infinite determinants, due to Hill and von Koch, and expounded 
in Chapter Vili. 

Without entering upon details, it may be said briefly that 
many of the properties of linear differential equations having 
algebraic coefficients can be treated by processes that, except as 
to greater complexity in the mere analysis, are the same as for 
equations with uniform coefficients. It therefore seems un- 
necessary to discuss them at any length, as they would lead to 
what is substantially a repetition of a discussion already effected 
for less complicated equations. 

■ Seo g 127. 

t Creiie, t. lxsy (1873), pp. 177—223. 

J CrelU, t. LKXKiii (1877), pp. 185-310. 

§ Acta Math., t. iv (1881), pp. 208 et aeq. 



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154.] EXAMPLE 483 

A systematic discussinn of equations having algebraic coeffi- 
cients and development of many of their chaiacteristie properties 
will be found in a series of memoirs by Thom6*. 
Ex. 1. Consider the equation 



where the variable y is dciined by the relation 

' +y=i 

and a, a, b, c are Luiistant^ 

The position, at infiiiitj i& a singuliiitj of the difleiential equation in 
©aoh of the two sheets ot tlie Eiemann suifice The integrals aie regular 
in that vicinity in one oheet ^nd the exponents to which thej belong ire the 
roots of 

provided o + 6i is not aero ; but, if a + bi=0, the integrals are irregular at 
infinity in that sheet. Similarly, they are r^ular in the vioinity of infinity 
in the other sheet, and the exponents to which they belong are the roots of 

'<'+'>+5^-K).-''' 

provided a — bi is not zero; but, if a — iii = 0, the integrals are irregular at 
infinity in that sheet. 

The other singularities of the equation are given by 
ax+by-VC=0\ 

When these are distinct hma one another, let them be denoted by ; =cos fl, 
y = sm6; x=aoa(j>, y=3mi^ The integrals are regular in the viuimty of 
each position; and the leajectue mdiciil equations are 

''"■^" + („ S°c.t«/ -° 

'''-"■^ (»-i°c.« <-°' 
When the two singularities coincide, let the common position be denoted by 
a:=c<m\lf, y — sirt'jr ; and then 

In the vicinity, we have 

ar=cos-J,+a 

, = .int-|cot^-i_^!^ + jC^-..., 

* Crelle, I. cxv (1895). pp, 33—33, 119—149 ; ib., t. csix (1898), pp. 131—147; 
ib., t. oixr (1900), pp. 1—39; ib., t. cslsii (1900), pp. 1—39. 



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484 EXAMPLES OF EQUATIONS HAVING [154. 

so that the equation is 

The integrals are not r^ular ; but the equation may have one normal 
integral, and can even have two normal integrals, of the type 

aUin V ! 

■where / is ft polynomial in ^. The forms, and the conditions necessary to 
significance, can be obtained as in §§ 86 — 87. 

Ex. 3. Discuss in the same way the singularities of the same differential 
equation, when the irrational quantity y is given by the respective relations 
(i) ^+f = \, 
(ii) /-4i^-yja^-^3. 
Ex. 3. Let w, and a, denote a fundamental system, of the equation in 
Es. 1, for 3' = (1— a^)^ ; and let «, and r^ denote a fundamental system of the 
same equation for y—-(l-a^)*. Shew tliat the linear equation of the fourth 
order, which has Mj, it^, Vj, v^ as its integrals, has rational functiona of x for 
its coeffioieiits ; and obtain them. 

Ex. 4. The equation 

has its primitive in tlie form 

It is natural to inquire whether an equation 

£^-^,. 

can have an integral of the type 

where nr (a:, y) is a rational function of a: and y. A general method for such 
aa inquiry has been given by Appell*, thoi^h it is not can'ied to a complete 
issue as regards detail ; it will be sufficiently illustrated by means of the 
equation 

iP"W_ a x-^^y _ 

where a?-^y^ = \, it being required to find under what conditions, if any, the 
equation can have an integral of the form 



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154.] ALGEBRAIC COEFFICIENTS 

where nr (,«, y) m s, rational function of x and y. Since 



We assume that each of the quantities a+flt, a+hi, is different from nero. 
By adopting the method in the precediug Ex. 1, the integrals of the equation 
in w are easily aeen to b« regular in the vioinity of a; = co , so that they have 
the form 

where ni-] is a holomorphic function for large values of x, not vanishing 
when :e=co ; and thus 



Substituting in the equation for or, we have 






Now the infinities of w are included among the points 

(i) :e= tc , which has juafc been considered ; there are two jwsaible 

values of X in each sheet : 
(ii) y = 0, with .r = l, ^=-1, which are the branch-points of the 

surface : 
(iii) ax-\-hy = <i, in each sheet. 
Moreover, the zeros of w are uuknowti from the differential equation : but 
they must bo considered, becaiise each of thetu gives a pole of la. Let such 



the number of suoh points being unknowiL All these points, whether 
infinities of w or zeros of w, can be singularities of -m. 

As regards the branch-points (ii), we may take 

y = 'i, ^=I-iiH..., 

in the vicinity of 1, 0, where jj is small ; and then 

so far as the governing term in ct is concerned. If tiiis be 



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486 EQUATION OF SECOND ORDEIl [154. 

where ft>0, then 

n+2=2n, A'^+nA=0. 

Thus n=2; and we can have ^= — 2, or ^ = 0, as possible values, 

Similarij for the vicinity of — 1, 0. 

Next, at the two points (iii), where as+iri/=0, we have 

^^^sinvf', y = coii\jr-, a tan 'jr = -- b. 
Then, in the vicinity, wo take 

,K=^ain.;^ + |, y = oosi|.-^Uni^+.,., 

Thus the equation is 

d^ (.1.-8..) »1 

de*' («■+»•)' e' 

su far as the gaverning term in cf is concerned. If this governing term be 
T .(»■+» ) 1 

' '- (»■+!.•)■ ■ 
Thus there are two possible values of rr at each of the two points. 

Lastly, as regards a point such as ,i;=/in the set (iv), it is easy to see that, 
if the governing term in ot be 

B 

(^ -/')•" 

2n = n + ^, B'i = nB ; 

that 18, 11=1, and either B = \, 5=0, are possible values. This holds for 
every such point a:=f and in each sheet. 

Our required function luix, y), if it exists, is to be a rational function of 
X and j/t and we have obtained all the singularities that, in any circumstances, 
it might possess. We accordingly must take some combination of the possible 
iiifinitiea, which are 

j; = oo , with any of the values of X, 

^-±l,y=0, with either A = -2,oi A=0, 

a3: + by = Qi, with any of the values of a, 

^=/, with 5=1, or 5-0. 



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154.] HAVING ALGEBRAIC COEFFICIENTS 487 

A possible form is clearly 

a 

where (7 is a constant. We have (if this be admissible) 

a + bi ' 
from the first of the possible infinities : we take A = from the second ; then 

<-'='°"?' 

from the third ; and we take 3=0 from t!ie fourth. Hence we muat have 
a-bi 

for some possible values of X and of a- : that is, 

-{-gyp)'=^T-{'-<.->s>?'--«}'} 

the signs being at oiir disposal. Thi.s leads to a single value of ft via. 

and the condition ia satisfied by taking the negative sign on both siiies. 
We then have 

so that, with the above value of |3, an integral of the equation 

d^^ 2/iax-i-byy 
is given by 

Actual evaluation of the integral in the exponential can easily be effected. 

se, it would have been possible to discviss the particular equation 



by taking 



=l+(i' ^~i+(a' 



with ( as the new independent variable ; for the algebraic relation is of genua 
zero, and therefore* the variables can be expressed as rational functions of a 
new parameter. The new form of equation would then have uniform coeffi- 
cients. But the foregoing method, that has been adopted, ia possible for an 
equation ijr {x, y) = of any genua. 

* T. F., g 247. 



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INTRODUCTION OF [155, 



Association with Auxomokphic Functions. 

165. It is manifest that some of the complexity in the 
analysis associated with the construction of integrals, either in 
general or in the vicinity of particular points, would be removed, 
if the equation could be changed so that, in its new form, its 
coefficients are uniform functions of the independent variable. 
This change would be secured, if both the variables x and y in 
the relation 

were expressed as uniform functions of a new variable z. 

Now it is known* that, when the genus of this relation is zero, 
both X and y can be expressed aa rational functions of a new 
variable z, which itself is a rational function of x and y. moreover, 
the expressions contain (explicitly or inaplicitly) three arbitrary 
parameters, which may be used to simplify the form of the 
resulting equation. Againf, when the genus of the relation is 
unity, both a: and y can be expressed as uniform doubly -periodic 
functions of a new variable z, while tg(z) and ^ {z) are rational 
functions of x and y ; moreover, the expressions contain (explicitly 
or implicitly) one arbitrary parameter, which again may be used 
to simplify the form of the resulting equation. And, in each case, 
definite processes are known by which the formal expressions of x 
and y, in terms of the new variable, can actually be obtained. 

When the genus of the algebraical relation 

^{!e,y) = ^ 

is greater than unity, a corresponding transformation is possible 
by means of automorphic functions : not merely so, but such a 
transformation can be efl'ected in an unlimited number of ways. 
Further, it is possible to choose transformations that simplify the 
properties of the integrals of the diflerential equations to which 
they are applied. But, down to the present time, the instances 
in which the complete formal expressions of x and y have been 
obtained, and the application to the differential equations has 
been made, are comparatively rare. The results that have been 

* T. F., % 247. 



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155.] AUTOMORPHIC FUNCTIONS 489 

established are of the nature of existence- theorems. It is true 
that indications for the construction of formal expressions are 
given ; but the detailed analysis required to carry out the indica- 
tions is of so elaborate a charactet that it may fairly be said to be 
incomplete. The subject presents great, if difficult, opportunities 
for research in its present stage. 

A brief account, based mainly on the work* of Poincar^, is ail 
that will be given here. References to the investigations of Klein 
and others in the region of automorphic functions will be found 
elsewheref. 

The main properties of infinite discontinuous groups and of 
functions, which are automorphic for the substitutions of the 
groups, will be regarded as known. It is convenient to associate 
with any group a region of variation of the variable which is a 
fundamental region ; and for the sake of simplicity in the following 
explanations, it will be assumed that this region is such that, 
when the substitutions are applied to it in turn, the whole plane 
is covered once, and once only. Further, also for the sake of 
simplicity, it will be assumed that the axis of real quantities in 
the plane is conserved by the sufetitutions of the group. There 
are corresponding investigations, which establish the results when 
these assumptions are not made; but, as already indicated, the 
results are mainly of the nature of existence-theorems and cannot 
be regarded as possessing any final form, so that the kind of con- 
sideration adduced will be sufficiently illustrated by dealing with 
the simplest cases. In order to deal with the most general cases, 
it is nece^ary to utilise the theory of automorphic functions in all 
its generality ; yet the subject still is merely in a stage of growth, 
being far from its complete development^, 

156. It is known § that, if x and y be two uniform 
functions of a variable z, which are automorphic for an infinite 

* This work ia beet esponnded in his five valuable memoirs in Acta Matheviatica, 
t. 1 (1882), pp. 1—63, 193—294. ib., t. in (1883), pp. 49—99, ib.. t. iv (1884), 
pp. 201—312, ib., t. V (1884), pp. 209—278. 

+ T. F., chapters isi, ixii. 

t The most oonseeotive account of the subject is to be found in Frieke und 
Klein's VorUmngen ii. d. TkeorU d, mttmiiorphen Functionen (Leipeig, Teahner; 
vol, I, 1897; vol. 11, part i, 1901). 

§ T. F., § S09. 



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'490 AUTOMORPHIC [156. 

discontinuous group of substitutions effected on z, then some 
algebraic relation 

+(«,j)-o 

subsists between them. Conversely, if this algebraic equation be 
given, it is desirable to express the variables x and y as uniform 
automorphic functions of a new variable z. For this purpose, we 
note that for general values of x, the variable i/ is a uniform 
analytic function* of m ; but there are special values of x, being 
the branch -point 8, at and near which y ceases to be uniform. 
Now suppose that x can be expressed as a uniform automorphic 
function of s, say 

the fundamental polygon being such that the branch-point values 
of X cori'espond to its comers (or to some of them), which include 
all the essential singularities of the uniform function /(s). Then, 
when substitution is made in the above relation, it becomes an 
equation defining ?/ as a function of ^ ; so long as z varies within 
the poiygona! region, y does not approach those values where it 
ceases to be uniform, for they are given only by the corners of the 
polygon. Hence y becomes a uniform function-t* of s ; and as a; is 
automorphic for the group of the polygon, it is at once seen that 
y also is automorphic for that group. 

Further, suppose that at the same time there is given a linear 
differential equation of any order, in which the coefficients are 
rational functions of sc and y. In addition to the branch-points 
which may be singularities of the equation, it naay have a limited 
number of other singularities. Let such a singularity be x = a, 
y — h, where of course 'if {a, b) = 0: for the moment, the question 
of the regularity of the integrals in the vicinity is not raised. If 
the polygon is constructed, so that x = a corresponds to one of its 
corners which is an essential singularity of the group, then that 
corner is an essential singularity of /(a). Hence, when the 
differential equation is transformed so that z becomes the in- 
dependent variable, the original singularities no longer occur so 
long as 3 is restricted to variation within the fundamental polygon : 
they can occur only for the special values at the corresponding 
If, further, the function f(s) is such that no special 



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156.] FUNCTIONS 491 

singularities for values of s arc introduced for values of w that are 
ordinary points of the equation, which will be the case ii f'{z) 
does not vanish within the polygon, then all the values of z within 
the polygon are ordinary poiuts of the equation, and all the 
integrals are synectic everywhere within the polygon. The 
singularities have been transferred to the boundary of the s-region; 
and thus the variables x and y, as well as all the integrals of the 
given linear differential equation which has rational functions of x 
and y for its coefficients, can be expressed as uniform functions of 
z within the region of its variation. 



AuTOMouPHic Functions and Conformal Bepbesenta.tion. 

157. The relation between the variable s and the function 
x=f{s) can be considered in two different ways, the analytical 
expression of the significance being the same for the two ways. 

In the first place, the relation can be regarded as one of 
conformal representation. Assuming for the sake of simplicity 
that all the singular values of x are real, consider the problem* 
of representing the upper half of the 3;-plane bounded by the axis 
of real quantities conformally upon a polygon in the s-plane, 
bounded by circular arcs and having m sides : this conformal 
representation is known to he possible. If its expression be 

then f'{z) must not become zero or infinite anywhere within the 
polygon, that is, for any finite values of x; for otherwise, the 
magnification would be zero or infinite there, a result that is 
excluded save at possible singularities on the boundary. 

It is manifest that the representation remains substantially 
the same, if the s-plane be subjected to any homograpiiic trans- 
formation 

where a'd' — h'd — \\ for this will merely change the polygon 
bounded by circular arcs into another polygon similarly bounded. 

* T. F., % 271. 



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492 CONFORMAL REPRESENTATION AND [157. 

Hence, in constructing the function for the conformal representa- 
tion, account nmst be taken of this possibility; and therefore, as 









{'. 


4 = B,«'l, 






where 


{'.'] 


is the Schwar 


ziai 


1 derivative, 


we 


construct thi 


tion {z, 


4 


We hare' 














k«i=i5; 


1- 


^ + s^ 


o° 


. il(x), 




(S- 


-0)- + *:.- 



say, where the summation on the right-hand side extends over all 
the singular values a of a;; the interna! angle of the s-polygon at 
the corner homologous with a is a-rr, and the coefficients A^ are 
real quantities. If oo is an ordinary value of x, so that no angular 
point of the polygon is its homologuo, then 

= tA„ 

= XaA + lS(l-aa 

= 2aU»+Sa(l~a^). 

li' oD is a singular value of x, whicii has an angular point of 
the polygon as its homologue, with the internal angle equal to ictt, 
then 

the summations being over all the finite singular values of x. 

The number of constants is sufficient for the representation. 
In the case when oo is not the homologue of an angular point of 
the polygon, we have m constants a,, m constants a, and m con- 
stants A„, subjected to three relations as above; as all these 
constants are real, there are 3m — 3 independent constants. 
But, if 

_ a'-X + b" 
^ c"'X + d" • 

where a"d" ~ ^"c" = 1 and the constants a", b", c", d" are real, 
then the upper half of the ic-plane is transformed into itself; 
hence the m constants a are effectively only m — 3 in number, 
and thus the constants in / («) are equivalent to 3m — 6 inde- 
pendent constants, which can be used to make a solution determ- 

■ The whole iovestigation is due to Schwarn; see T. F., g 371. 



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157.] AUTOMORPHIC FUNCTIONS 493 

inate. On the other hand, to determine the polygon, 3m constants 
are needed, viz. two coordinates for each of the m corners and a 
radius for each arc : but these are subject to a reduction by 6, for 
the representation is determinate subject to a transformation 

c'^+d" 

where a'd' — b'c' = 1, and the constants a', b', c', d' are complex, so 
that there are six real parameters undetermined. The number of 
available constants is therefore sufficient for the number of condi- 
tions that must be s 



In the case when =0 is the homologue of an angular point, we 
have ni — 1 constants a, m constants a, and m constants A„, sub- 
jected to two relations as above; as ail these constants are real, 
they are equivalent to 3m — 3 independent constants. The re- 
mainder of the argument is the same as before ; and we infer that 
the number of constants is sufficient to satisfy the number of 
conditions for the conform al representation. 

It need hardly be pointed out that, thus far, the polygon 
bounded by circular arcs is any polygon whatever; it has been 
taken arbitrarily, and it does not necessarily satisfy the conditions 
of being a fundamental region suited for the construction of auto- 
morphic functions. 

158. That polygons can be drawn in the a-plane, suited to 
the construction of autoraorphic functions in connection with a 
given algebraic relation i/r (x, y) = 0, may be seen as follows. For 
simplicity, let the polygon be of the first family*, and let it 
have 2n. edges arranged in n conjugate pairs ; and suppose that q 
is the number of cycles of its corners, each cycle being closed. 
The genus p of the group is given by 

2p = m -i- 1 - g. 

When the surface included by the polygon is deformed and 
stretchetl in such a manner that conjugate edges are made to 
coincide by the coincidence of homologous points, then for each 
cycle in the polygon there is a single position on the closed 

" T. F., %% 203, 293. 



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494 FUNDAMENTAL REGION [158. 

surface obtained by the deformation. This closed surface corre- 
sponds* to the Riemann surface for the equation 

^}r {x, y) = 0, 
which also is of genus p ; and thus there are q positions on the 
surface. ea<;h associated with one of the q cycles. Each such 
position requires a couple of real parameters to define it; and 
thus we have 2§ real parameters. Equations, which are biration- 
ally transformable into one another, are not regarded as inde- 
pendent r and therefore the effective number of constants in 
^ {^' y) = to be taken into account is 'ip — 3, being the number"!" 
of class-moduli which are invariantive under birational transform- 
ation. Each of these is complex, so that the number of real 
parameters thus arising is 6p — f). We therefore have to provide 
for 6p — 6 + 25 real parameters, by means of the polygon. 

In order that the polygon may be properly associated with a 
Fuchaian group, it must satisfy certain conditions. Its sides must 
be arcs of circles, the centres of which lie in the axis of real 
quantities. As it has 2n sides, we therefore require 2ji centres on 
that axis and tn radii, making 4ft real quantities in all ; but three 
of the centres may be taken arbitrarily, for the polygon now 
under consideration is substantially unaffected by a transforma- 
tion 

V ' cz + dl ' 
where a, b, c, d are real ; so that the total number of real quanti- 
ties necessary is effectively 4« — 3. They are, however, not suffi- 
cient of themselves to specify an appropriate polygon 1 for 
conjugate sides must be congruent, a property that imposes one 
condition for each pair of edges, and therefore n conditions in all : 
and the sum of the angles in a cycle must be a submultiple of 2-n; 
so that q conditions in all are thus imposed. Hence the total 
number of real quantities necessary is 

= 4m — S — n — q 

= Sn.- ^~q 

= 6p~6 + 2q, 
in effect, the same as the number of real parameters given. 



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159.] FUCHSIAN EQUATIONS 495 

AUTOMORPHic Functions and Linear Equations oB' the 
Second Order : Fuchsian Equations. 

159. In the second place, the variable s, and the automorphic 
functions x and y, can be associated with a Unear differential 
equation of the second order. Let 



then it is easy to verify that 

v^ da? «s &3? IV ^ ' 

where (*', z\ is the Schwarzian derivative of as with regard to z, 
and 1^ = dxjd,z. It is a known property* that, if *■ is an auto- 
morphic function of z, tben the function 



is automorphic for the same group; hence it can ] 
rationally in terms of a; and y, where 

Denotiog its value by — 21, where / is a rational function of 
X and y, which may be a rational function of x alone, we have 
V] and Vj as linearly independent integrals of the equation 

,-„ + /jf = ; 

the quantity z is the quotient of the two integrals. 

The analytical relation is effectively the same as before ; 
for if 

{z,x] = 2I, 

we knowf that z is the quotient of two integrals of 



n Difereiitial Kquatioi 



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496 AUTOMOKPHIC FUNCTIONS AND [159. 

Moreover, 

90 that the results agree in form. The difference is that, regard- 
ing the relation as a problem of conformal representation, we 
have been able to calculate the value of / in gi'eater detail than 
in the alternative mode of regarding the relation: but the con- 
siderations adduced in connection with the differential equation 
have been of only the most general character, and have not 
permitted any discussion of the form of /. 

When an equation of the form 

da? 
is given, where / is a rational function of x, or a rational function 
of two variables x and y, connected by an algebraic equation 

t(!l!.j)-0, 

it may happen that x and y are uniform fuQctions of z, the 
quotient of two integrals of the differentia! equation. But these 
uniform functions are not necessaiily, nor even generally, auto- 
morphic for a group of substitutions of s. Judging from the result 
of the consideration of the question as a problem of conformal 
representation, we should be led to expect that the constants, 
which survive in / after the conditions for uniformity are satisfied, 
might be determinable so that the uniform functions of z are 
automorphic. When this determination is effected, the equation 
is called* Fuchsian by Poincar^j if the group be Fuchsian. 

160. We proceed to consider more particularly the properties 
of the equation 

,-„-|-/)i = fl, 

in relation to the qiiotient of its integrals. Let jc = a, y = h be a 
singularity of the equation, where ■^{a,b) = 0; and let 

Limit [{x — ayi]it=a = p, 
so that the indicial equation for a is 

n(n-l) + p = 0. 
' Acta Math., t. IV, p. 323. 



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160.] FUCHSIAN EQUATIONS 497 

Let n, and n^ be its roots, when they are unequal ; then two 
integrals of the equation are of the form 

and BO 

If atr be the internal angle of the jr-polygon at the angular point 
homologous with a, we must have 



and therefore 
that is, 
so that 



— 4p = 
1-a 



the remaining terms being of index higher than — 2. 

This is valid, if a is not zero. When a is zero, ao that «i = % 
and therefore p — i, the integrals of the equation are 

jjj = (3;-([)«i[[l + ...} log (fl! -«) + powers oix — a], 
and so, in the immediate vicinity of a, we have 
2 = — = log («! — »)+ powers ; 
and then 

the remaining terms again being of index higher than — 2. 

The quantity a, in terms of which the leading fraction in Z is 
expressed, depends upon the character of the singularity at (a, b). 
If the latter denote a singular combination of values for the 
equation 

then it is known* that the variables x and y can be expressed in 
the form 

' T. F., § 97. 
T. iv. 32 



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498 FUCHSIAN [160. 

where S(^) is a regular function of f. which does not vanish when 
f = 0, and the expressions are valid in the immediate vicinity of 
the position. Let r be the least common multiple of p and q, and 
write 

then in that vicinity, we have 

{x-ar = z+..., 

SO that both x and y are uniform functions of s in the vicinity. 

The commonest instance occurs, when {a, b) is a simple branch- 
point ; we then have 

so that a = 4, 

If (a, b) he a singularity of some given differential equation of 
any order, say 

where i/r (x, y) = 0, three cases arise. 

Firstly, let all the integrals be free from logarithms, and let all 
the exponents to which the members of a fundamental system of 
integi'als (supposed regular) belong be commensurable ; then they 
are integer multiples of a quantity k~^, and we take 

..l. (.-„, = ...... 

In that case, any integral is of the form 

= (a; — o)* R(a; — a) 

-''RC), 

SO that the integrals of the equation, as wel! as the variables 
w and y, become uniform functions of z in the vicinity of 2 = 0. 

Secondly, let the integrals (still supposed regular) of the 
fundamental system belong to exponents some of which at least 
are not commensurable quantities. We take 
ir = log(a!-a)-|-powers; 



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becomes 



integral of the form 

ix~arR(a:-a), 



i.e., a uniform function of s, valid for large values of \z\ : and this 
uniformity is maintained whether /j, is commensurable or not. 

Thirdly, let x = a be an essential singularity of one or more of 
the integrals, supposed irregular there. As in the laat case, we 
take 

2 = log (ai ~a) + powers ; 

the integral may or may not become uniform for large values 
of U|. 

In the last two cases, if the expression for a: in terms of s, say 

be automorphic, then 2 = os is an essential singularity of the 
function f{s) ; and then, when z varies within the polygonal 
region, w does not approach the value a for which the integrals of 
the equation cease to be regular. Within the region, the integrals 
are unifoi-m. It is to be noted that the relation, adopted in the 
second case and the third case, woufd be effective in the first case 
also, so far as securing uniformity ; but the converse does not 
hold. The relation which, as seen above, corresponds to the 
vicinity of an angular point of the polygon where the sides touch, 
is the most generally applicable of all : the form of relation, corre- 
sponding to the first case, is applicable only under the somewhat 
restricted conditions of that case. 

161. These conditions and limitations affect the quantity / 
in the equation 



for they determine the leading coefficient in its expansion near 
any of its poles ; but, in general, they do not determine / com- 
pletely. On the other hand, we so far have only secured the 
uniformity in character of the functional expression of x in terms 
of z: the automorphic property of the functional expression has 
not been secured. The latter is effected by the proper assign- 
ment of the remaining parameters in /. 



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500 CONSTRUCTION OF [161, 

Aa a special instance, take the case in which the genus of the 
group and of the permanent equatii>n is zero ; so that, if the 
polygon has 2m edges, the number of cycles q is given by 

, = » + !. 
Taking the angulai' points in order as A„ A^, ...,A^a, and making 
the sides 

A,A, I A,A, ] ] A,^,A^ ] An ^„+,) 
A,aJ' A,„A^J---y A„+,A„J' A„+,A^J- 
to be conjugate pairs, the necessary n+1 cycloa are 

A,] A„A,^; A,-4™-i; ■-; A,„A„+,; A„+„ 
To define the polygon of 2n circular arcs, which have their 
centres on the axis of real quantities, we require the 4in coordi- 
nates of the angular points ; but these effectively are only 4k — 3 
quantities, because the ^-plane is determinate, subject only to a 
transformation 

/ as + b\ 
V ' cz + d)' 
where a, b, c, d are real. In each cycle, the sum of the angles is 
a submultiple of Stt : so that n + 1 conditions are thus imposed. 
Again, the edges in a conjugate pair must be congruent; so that 
n furthei' conditions are thus imposed. Accordingly, there remain 
2ji — 4 real independent constants to determine the polygon. 

The polygon thus dofcermined defines a Fuchsian function; as 
the genus is zero, every function can be expressed rationally in 
terms of x, so that the equation for v (leading to s, as the quotient 
of two integrals) is 

-3- +lv=0, 

ax' 
where / is a rational function of x. Corresponding to the n + 1 
cycles, there are w + 1 values of x ; let these be 



Let OtTt be the sum of the internal angles of the z-polygon corre- 
sponding to (X,, so that Uj. is the reciprocal of an integer; and 
take 0,1.^1 to be the quantity a for co . Then in the vicinity of a^, 
we have 



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161.] FUCHSIAN EQUATIONS 

for each of the values of r. Thus, if we wribe 

and remember that I is a, rational function of x, we have 



«'«''=["]:.„■ 



for r=l, ...,n. In order to satisfy the condition for « = !», 
6(x) must be of order 2« — 2, and 

aw- 1(1 -<■'«)«■"-+•... 

The number of coefficients in G (x) is 2m — 1 ; but the coefficient 
of the highest power is known, and there are n relations among 
the rest, owing to the conditions at CTi, ..., a„; lience there remain 
w — 2 coefficients independent of one another. Each of these is 
complex in general, so that they are effectively equivalent to 
2w — 4 real constants. Assuming that the quantities Oj, .,., «« 
are known, it is to be expected that the 2« — 4 conditions for the 
polygon determine these 2n — 4 real constants. 

; and we may take Os — O, 





In the simplest 


casc, \ 


ve have n 


,= 


<h-- 


= 1, so that 










/ = i 


a? 


■-i(^ 


I)' 


The conditions for . 


X = 'K, 


give 










f 


+ 




Hi- 


".=) + 


J(l-< 


) + 



= i (!-«.■). 

where a,, Kj, eta are the reciprocals of integers; the quantity / 
then is the invariant of the hypergeometric series. 

162. As another illustration, which may be treated somewhat 
differently, consider the equation 

1/' = a; (1 - ic) (1 - Qx\ 

where c is a real constant less than unity ; and write 

ac = 1, 



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502 EXAMPLE OF A [162. 

SO that a is a, real constant greater than unity. Here, the points 
ic= 0, 1, tt, X are ea«h of them singular; and the value of a is ^ 
for each of them. Consequently, 

, l(i-i), Hi-i) , i(i-i) ,^ B c , 
'- -„'' '* (p-iy +(«-»,)■ + ». + ^- ! + «-«• 

and the conditions for « = co give 

One constant in / is left undetermined by these conditions ; thus 

^="+(J^ + (S?S)'-^l)(»!-«)' 

say, where \ is the undetermined constant. U is possible to 
determine \, so that a: is a Fuchsian function of z, where z is 
the quotient of two solutions of the equation 

-- + /d = 0. 
dx' 

As regai-ds this Fuchsian function, its polygon may be obtained 
simply aa follows. We take four points A, B, G, D in the 2-plane 
to be the homologues of 0, 1, a, x ; owing to the value of a, which 
is ^ in each case, the internal angies of the polygon must each be 
\ir. We make the edges AB, CD conjugate, and likewise the 
edges BG, DA ; and then there is a single cycle, ADGB, the sum 
of the angles in which is %-Tr. With the former notation, we thus 
have 2 = 1, « = 2 ; so that 

2p = 2 + 1 - 1 = 2, 

and therefore p = 1, as should be the case. Further, the sum of the 
angles of a curvilinear triangle, entirely on one side of the real 
axis, is less than tt, when the centres of the circular arcs he on 
the real axis : so that, if our polygon be curvilinear, the sum of its 
angles would be less than 27r (for it could be made up of two 
triangles), whereas the sum is actually 2Tr. Hence the polygon 
can only be a rectangle, and the Fuchsian functions are doubly- 
periodic. We therefore take 

a: = sn' ^, y^snzcnz dn z, 



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62.] FUCHSIAN EQUATION 

s is manifestly permissible ; and then 



dx '2.y 2cM(a;-l)i(a.'-(i.)i' 
which leads to 

''■■"I • L^ + (a, -1)' + (»■-(.)'] ^x{x^\){„-a) 

-21, 

so that we have 

X._J(„ + 1). 

This value of X renders x (and so y) a Fuehsian funetioa of the 
quotient of two solutions of the equation 



As regards the integrals of this equation, the indicial equation 
of« = Ois 

/>(p-i) + iV-». 

SO that p = ^, p = I- Denoting by Vi and v^ the integrals that 
belong to J and J respectively, we have 



! + §-:>■+... 
= ^-j;i . 



-i'-iii+c>e'+... 

= en" f, 
after the earlier analysis. 

Similarly, in the vicinity of « = 1, wo find integrals 



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604 




EXAMPLE OF A 


and then, taking 








,^^. 


we find 






Now 






»-l- 


-cn= 


t 


= 


-(1- 




- 


-0- 


-«)(f-2)' + i(l-2»)(l-«)(f-^)' 


so that 




i, = (o-l)HI;-K). 


Hence 




!:-<-')' e-^)' 


so that, as 






where AB- 


-BC. 


- 1, because 


we have 










Again, i] 


nthe 


vicinity of « = a, we find integrals 



[162. 



cr,-(^-o)i|i + 2A;^j''^(^-») + ...|, 



and then, takini 



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162.] FUCHSIAS EQUATION 

we find 

Also 

ic — cr = sn' c 

c 

= — dn^ f 

^—-{K-K-iKj + }S^—^-—''hK-K-iKy^..., 

so that 

Proceeding as before, this leads to the relations 

Lastly, for large values of x, we have 

F,-«i|l-A(l+o)i+...l, 

r,-^|l-l(l + o)i+...}; 
and then, taking 

w. 



we find 
Now 



--6--|<l+a)f.'+.... 

1 1 
«-sn-r 

=csL"(f-iin 

= c(r-iir7-ic-(r-!'Jr')'(i+<»)4 



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506 INTEGRALS EXPRESSED AS [162. 

SO that 

Proceeding as before, this leads to the relations 
If, = ci (v^ - iK\) \ 
Wy = o-^v, ] ■ 

The relations, in fact, have enabled us to construct the expressions 
for each fundamental system in terms of the first and, therefore by 
inference, in terms of every other. 

Ex. 1. Discuss in the same way the Fuchsian differential equation 
1 Ai f J 1 _\ 1 1^ 

connected with the eqiiation 

Ex. 2. Shew that, if 
whei'e p denotes Weieratrass's elliptic function, 

'•■ ■•'-•Ls^^lJ^ + (J:^' * (i:^'J"*(i-«,)(»-«,r(i^"S ■ 

and discuss the significance of the integral relation in regard to its paeudo- 
automorphic character for the equation 

Es:. 3. 4 f d e tal J hg x the s-plane is composed of two semi- 
circles, one upo a d am t i tl c real axis for values of s correaponding to 
values of a equal to and 1 the other upon a similar diameter for values of 
a: equal to 1 ■uid u (wl e e > 1) and of two straight lines drawn, through 
points corresj ondii^ to and a, perpendicular to the axis of real quantities. 
Prove that tlie subsidiary equation of the second order, for the construction 
of X as an automorphic function of the quotient of two of its integrals, is 

id,'- 'L"'^^ (»-!)■ + (i-o)>J^^»(«-l)(»-o)' 
where the constant jj is to be properly determined. 

AUTOMORI'HIC BY'NCTIONS USED TO MAKE THE InTEGRALH 

OF ANY Equation Uniform. 

163. If, for any given equation, there is only one singularity, 
it can be made to lie at the origin. 

In order to obtain a variable s, in terms of which the integrals 
of the given equation can be expressed uniformly, we construct an 



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163.] UNIFORM FUNCTIONS .507 

equation of the second order which has i« = for a singularity, 
of such a form that the indicial ecjuation for x = has equal 
roots (§ 160). This auxiliary equation may have other singulari- 
ties, but otherwise it may be kept as simple as possible. Such 
an equation is 



the indicial ecjuation for »: = is 

e{d-l)=X. 
90 that \~ — I if it has equal roots. Thus the equation is 

Two integrals are given by 

Vi = x^, fla = 3^ log « ; 

thU3 



which is the new independent variable. 

An equation of the kind indicated ia {§ 45, Ex. 6) 

when the variable ia changed from :>: to s, where j; = e', the equation hccomea 

The integrals are synectic for all finite values of 3. 

164, When a given differential equation has two singularities, 
a homographic transformation can be applied so as to fix them at 

X = 0. iK=l. 

To obtain a variable z in terms of which the integrals of the 
given equation can be expressed uniformly, we construct an 
equation of the second order, having and 1 as its singularities 
and such that the respective indicial equations have repeated 
roots. An appropriate equation is 

d'v _ a + ^w 



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508 EXPRESSION OF INTEGKALS [164. 

The indicial equation for « = is 

p{p-l) = a, 
so that a= — ^ ; the indicial equation f or ic = 1 is 

p(p-l) = «+/3. 
SO that a + ^ = — ^, and therefore ,8 = 0, so that the equation is 

One integral is easily found to be 

and then z, the quotient of another integral by v,, is given by 

dz ^C ^ - 1 
dx jii' x(a!~l)' 

on particularising the constant C, which may be arbitrary. Thus 

gives a new variable g, Huch that the integrals oi' the given 
differential equation arc uniform functions of s. 



Thus let the equatio: 



>a2' = <'. 



which has :f=0 and x=l for real singularitiea ; it is easy to verify that 
a:=ro ia not a Hingularity but only an ordinary point for every integral. 
When the equation is transformed so that b is the indepeodent variable, it 
becomes 

the integrals of which clearly are uniform functions of s. 

165. When a given differential equation has three singulari- 
ties, a homograpfiic transformation can be used so as to fix them 
at a;=0, 1, oc . 

We may proceed in two ways. It may be possible to choose, 
as the fundamental region in the s-planc, a triangle, having 
circular arcs for its sides, and having Xtt, fi-rr, vtt for its internal 
angles at points which are the homologues of 0, oo , 1 respectively : 



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165.] AS UNIFORM FUNCTIONS 

X, fi,, V being the reciprocals of integers. Then the 
equation may be taken in the form 

i(^% iO^-A^) iOr''') , i^'-'-'-^"'-' 



which is the norma! form of the equation of the hypergeometric 
series with parameters a, jS, 7, where 

X' = a-7)<, ^■ = (a-«', „■ = (,- = -/3)-. 

The variable s may be taken as the quotient of two solutions of 
the subsidiary equation ; and so 



.fCa + l-T, ff + 1- 



") 



It is known* that x, thus defined, is a uniform automorphic 
function of s. 

This transformation will render uniform the integrals of a 
differential equation, which has no aiogularities except at 0, 1, 
00 , provided the integrals are regular in the vicinity of those 
singularities and belong to indices which are integer multiples of 
X, V, fi respectively. If these conditions are not satisfied, in 
particular, if the singularities are essential for the integrals, then 
we proceed by an alternative methoi^. 

We take a subsidiary equation having 0, 1, co for singularities, 
such that the indicial equation for each of them has equal roots. 
Let it be 

where a', ^3", y are to be chosen so that the indicial ecjuation for 
each of the singularities has equal roots. These equations are 

p(p-l) = a', <T(a--l) = a' + ^' + y', t(t + 1) = 7', 
so that 

"'--i, f>' = i. 7'--i, 

and thus the equation is 

• r. F., § 275. 



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510 GENERAL APPLICATION OF [165. 

The coefficient of v is the invariant of a hyp orgeo metric equation, 
of which the parameters are 

a = 0^l, 7 = 1; 

so that s, the quotient of two integrals v, is also the quotient of 
two integrals of the equation 



''diu 



-i» = 0. 



This is the equation of the quarter-periods in elliptic functions: 
BO that 

This relation effectively defines a; as a modular function* of z : 
the fundamental region is a curvilinear triangle. The function 
exists over the whole s-plane : the axis of real quantities is a line 
of essential singularity. 

Any differential equation, having a; = 0, 1, v> for all its singu- 
larities no matter what their character may he, can be transformed 
by the preceding relation so that a is the independent variable ; 
its integrals are then expressible as functions of z which are 
uniform over the whole of the e-plane, their essential singularities 
lying on the axis of real quantities. 

Ea:. A differential equation haa only tliree singularities at x=a, 6, e, 
such that the roots of the indicial equations of those points are int<^er 
multipleB of a, 8, y respectively, where a, ft y are reciprocals of integers. 
Shew that a variable, in terms of which the integrals can bo expressed as 
uniform functions, ia given by taking the quotient of two Riemann P-functioiis 
with the appropriate singularities aiid indices. 



AuTOMORPHic Functions applied to General Linear 
Equations of any Order. 

166. At the beginning of the preceding explanations and 
discussions, it was assumed {§ 157) that all the singular values 
of X are real. The assumption was then made for the sake of 
simplicity : it can be proved"!- to be unnecessary. 
* T. F., % 303. 

+ Poincare, Ada Math., I. iv, pp. 246—250. 



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1.66.] AUTOMORPHIC FUNCTIONS 511 

Firstly, let the singularities be constituted by a,, a^, ..., Om, 
all of which are real, and by c, which will be supposed complex. 
With these we shall associate Co, the conjugate of c ; and we write 

4>{x) = {x-c){a:-c„), 

a quadratic polynomial with real coefficients. Then all the 
quantities 

are real. Construct a fundamental region in the ^-plane, such 
that the foregoing m + 2 quantities are the homologues of the 
comere ; and let 

be the relation that gives the conformal representation of the 
region upon half the X-plane, so that F(s) is a Fuchsian faoction 
of 2. 

Consider the variable x, as defined by the equation 

So long as s remains within the fundamental region, a: is a 
uniform function of e; it could cease to be so, only if 

that is, if a! = ^c + ^c„, and then we should have 

y(») = .f(ic + }o.), 

which is not possible for values of z within the region. Also, 
J- is not zero for any value of s within the region; for then 
we should have 

which would make a zero magnification between the X-plane 
and the ^-region: this we know to be impossible for internal 
^-points. This uniform function x, whose derivative does not 
vanish within the polygon, cannot acqiiirc either of the values 
c or Co within the polygon, for then we should havo 

F{s) = 0. 



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512 AUTOMOBPHIC FUNCTIONS AND [166. 

which is possible only at a comer. Nor can it acquire any of 
the values til, eta, ..., a™ for points within the ;r- polygon : for 
at any such value, we have 

i*" («) = *(»). 

which again is possible only at a corner. 

Now since X = l'(3) is a relation that con formally represents 
the half X-plane upon a ^-polygon bounded by circular arcs (this 
polygon being otherwise apt for the construction of autornorphic 
functions), we have (§ 157) 

where ^ (X) is a rational function of X. But for any variables 
X and X, we have 

and therefore, in the present case, 

[z, «) = 2 (2a; ~ c - c„f f(af- cz -c^ + cc„) - 

= 2^(^), 

say, where '^V(x) is a rational function of ie. Hence s is the 
quotient of two integrals of the equation 

S + '^W'"- 

Now cc is known to be a uniform function of s ; it is therefore a 
Fuchsian function of z. And we have proved that, for values of z 
within the polygon, x cannot acquire any of the real values 
(ti, Oa, ..., (tm or either of the complex values c, d, and, further, 

that ^- does not vanish. 
as 

Secondly, to extend this result to the case, when x is not 

to acquire any one of any number of complex values for ir-points 

within the polygon, we adopt an inductive proof; we assume the 

result to hold when there are q — 1 pairs of conjugate complex 

values, and shall then prove it to hold when there are q pairs. It 

has been proved to hold, (i), when there are no complex values 

and, (ii), when there is a pair of conjugate complex values : it thus 

will be proved to hold generally. 



(2^ 



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166.] 



LINEAR EQUATIONS IN GENERAL 



513 



Suppose, then, that the given ic-singnlarities arc made up of a 
number m of real values a,,a.i, ...,fflm>and of a number of complex 
values. Let the latter be increased in number by associating with 
each complex value its conjugate complex, whenever that conjugate 
does not occur in the aggregate ; and let the increased aggregate 
be denoted by 

arranged in conjugate pairs. Write 



which is a polynomial of ( 
equation 



fi-ee 2q with real coefficients. The 



dx 



= 0, 



of degree 1q~l with real coeOicients, certainly p 
root ; its other roots, when not real, can be arranged in conjuj 
pairs, the number of pairs not being greater than g — 1. Let its 
roots be denoted by 

h, h, ..., Vi. 
an aggregate which contains not more than 5 — 1 conjugate pairs. 
In the series of quantities 

0; <f.(aO, ■-, 0(«-«); 0(M. ■■■. <^(6^-.); 
there are certainly m + 2 real quantities ; and there are not more 
than 5 — 1 conjugate pairs of complex quantities. According to 
our hypothesis, a Fuchsian function G{z) can be constructed, such 
that the foregoing m + 2 + 2 (5 - 1) quantities are the homologues 
of the comers of an appropriate fundamental region, and (?' (s) 
does not vanish within the region. Then, proceeding on the same 
lines as in the simpler case, we consider a variable .v, defined by 
the relation 

*W = (?{«). 

So long as s remains within the fundamental region, ic is a 
uniform function of 2 ; it could cease to be so, only if 

that is, it'a: = 6,, 6j or b;q-,, and then we should have 

e(«)-*(M, 4,(b,), .... or 4,(b„^o, 



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514 FL'CHSIAN EQUATIONS HATING [lfi6. 

which is not possible for values of s within the region. Also, 
-J- does not vanish for values of z within the region ; for otherwise 
we should have 

for such values, and this is known not to be the ease. Further, m, 
being a uniform function of z whose derivative does not vanish for 
values within tbe polygon, cannot acquire any of the values c,- or 
Cf, for r=\, ...,q, within the polygon; if it could, we should have 
(i (as) = there, and then 

i'W-o, 

which is possible only at a corner. Nor can it acquire any of the 
values a,, ..., Om for values of ^ within the polygon ; if it could, we 
should have 

F{z) = 4,{<h), (f){a,), ..., or ^(0> 

which again is possible only at a comer. 

Now since Y, = G(z), is an automorphic function, it follows* 
that 

which is equal to — (e, Y], also is an automorphic function. 
Consider the upper half of the F-plane. So far as the equation 
Y = G(z) is concerned, certain points on the upper side of the 
axis of real quantities are exceptional, not more than g — 1 in 
number ; these can be considered as excluded, and cuts drawn 
from them to singular points on the real axis. We then can 
regard this simply-connected and I'esolved half- plane as conformally 
represented upon the polygon by the equation F = G (s) ; hence-f 

where 5(F) is a rational function of F. But 

where </> (x) is a polynomial ; hence 

1^, «l-h Yli-^y + ir,-] 






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166.] ASSIGNED SINGULARITIES 515 

say, where 0(a;) is a rational function of x. Hence z is the 
quotient of two integrals of the equation 

Now a; is known to be a uniform function of z. It is therefore a 
Fuchsian function of z, which acquires the particular assigned 
values only at the corners of the fundamental region and nowhere 
within the region ; its derivative does not vanish anywhere within 
the region. 

The statement is thus established. 

167. The preceding explanations, outlines of proofs, and 
analysis, will give an indication of the kind of result to be 
obtained, and the kind of application to differential equations to 
be made. It will be recognised that such proofs as have been 
adduced are not entii'ely complete : thus, when a number of real 
constants is to be determined by the same number of equations, 
whether algebraical or transcendental, it would be necessary to 
shew that the constants, if determined in the precise number, are 
real. As, however, it was stated at the beginning of these sections 
that only an introductory sketch of the theory would be given, 
there will be no attempt to complete the preceding proofs : we 
shall be content with referring the student, for the long and com- 
plicated processes needed to establish even the existence of certain 
results without evaluating their exact form, to the classical memoirs 
by Poincar^, and to the treatise by Fricke and Klein, which have 
already been quoted*. It may be convenient to recount the 
most important and central results of Poincar^'s investigations, 
which have any application to the theory of linear differential 
equations. 

Let 

be a linear equation of order g, having rational functions of x and 
y for its coefficients, where y is defined in terms of a; by the 
algebraic equation 

' A memoi)- by E. T. Wliittaker, "On the eonneiion of algebraic funotions 
with automorphic functions," Fhil. Trami. (1899), pp. 1—32, may also be consulted. 



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516 POINCAEfi'S THEOREMS [167. 

tbis equation in w will be called the main equation. Let 



da:' 



H'.y) 



be another equation, in which Six, y) is a rational function of w 
and y ; it will be called the subsidiary equation, and its elements 
are entifeiy at our disposal. 

Let cc = o,^, 2/ = &^,bea singularity of the main equation. If all 
the integrals are regular at this singularity, if they ai-e i'ree from 
logarithms, and if they belong to exponents, which are commensur- 
able quantities {no two being equal), let fc~' {where k is an integer) 
be a quantity such that the exponents are integer multiples of ^'. 
We make a; = a^, y = b^,& singularity of the subsidiary equation. 
In the case of the indicated hypothesis as feo the integrals of the 
main equation, we make the difference of the two roots of the 
indioial equation of the subsidiary equation equal to k~^. In every 
other case, we make those two roots equal. This is to be effected 
for each of the singularities of the main equation. 

Thus the subsidiary equation is made to possess all the 
singularities of the main equation. It may have other singulari- 
ties also ; for each of them, the difference of the two roots of the 
corresponding indicial equation is made either zero or the re- 
ciprocal of an integer, at our own choice. By these conditions, the 
coefficient 6(x,y) will be partly determinate: but a number of 
i will remain undetermined. 



The effect of these conditions is, by the analysis of § 160, to 
make x and y uniform functions of e, where z is the quotient of 
two linearly independent integrals of the subsidiary equation ; 
and no further conditions for this purpose need be imposed upon 
the parameters, which may therefore be used to secure other 
properties of the uniform functions. The various forms of 6, 
corresponding to the various determinations of the parameters, 
determine a corresponding number of differential equations ; all 
of these are said to belong to the same type, which thus is 
characterised by the singularities and their indicial ( 



Poincar^ has proved a number of propositions connected with 
e results that can be obtained by the appropriate assignment 



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167.] ON AUTOMORPHIC FUNCTIONS 517 

of values to these parameters. Of these, the most important 

I. It is possible to assign a iinique set of values in such 
a way as to secure that x and y are Fuchsian functions of z, 
existing only within a fundamental circle. 

II. It is possible to assign sets of values, unlimited in 
number, in such a way in each case as to secure that x and 
y are Kleinian functions of z, existing over only part of the 
2-plane. 

III. It is possible to assign a unique set of values in such 
a way as to secure that x and y are Fuchsian functions or 
Kleinian functions of z, existing over the whole of the 3-plane. 

There are limiting cases when the Fuchsian function becomes 
doubly- periodic, or simply- periodic, or rational. 



PoiKCAR^'s Theorem that any Likbar Equation can be 

INTEGRATED BY" MEANS OF FuCHSUN AND ZeTAFUCHSIAN 

Functions. 

168. Consider now the integrals of the main differential 
equation, when they are expressed in tenns of the variable z. 
We shall assume that x and y have been determined as Fuchsian 
functions oi z, existing only within the fundamental circle. 

Near an ordinary point x^, y„, any integral w is a holomorphic 
function oix — x„; near such a point, ic is a holomorphic function 
of 2 — 2o ; so that w, when expressed as a function of 2, is a holo- 
morphic function of z. 

In the vicinity of a singularity (a, b), there are two cases to 
consider. If all the exponents to which the integrals belong are 
commensurable quantities, so that they are integer multiples of 
some proper fraction &-^ where k is an integer, and if the integrals 
are free from logarithms, then every integral is of the form 

«. = (x-»)'S(^-«). 



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518 POINCAR^'S THEOREM AS TO THE [168. 

where >5 is a holomorphic function of x — a,. As in § 160, we have 

SO that 

where T and li are holomorphic functions. Hence 

w = {z-cYG{z~c), 
where the function G is holomorphic in the vicinity of c. Thus w 
is a uniform function of 3 ; if /i is positive, then c is an ordinary 
point ; if ^ is negative, it is a pole. 

In all other cases, whether the integrals involve logarithms, or 
the exponents to which they belong are not all commensurable, or 
the singularity is one where some of the integrals, or even all the 
integrals, are irregular, the roots of the indicial equation for the 
subsidiary equation are equal. In consequence, the two circular 
arcs of any polygon touch, and thus the angular point is on the 
fundamental circle. As we consider the values of z within the 
fundamental circle, the character of the integral, when expressed 
as a function of z, does not arise for the point of the kind under 
consideration. 

It thus appears that, when z is restricted to lie within the 
fundamental circle of the Fuchsian functions which are the repre- 
sentative expressions of x and y, any integral of the main equation 
is a uniform function of z. When this uniform function has poles, 
it can be represented in the form 

where the zeros of G, (z) are the poles of the integral in unchanged 
multiplicity, and both 6 {s) and Qi {z) are holomorphic functions 
of z, within the fundamental circle. When the uniform function 
representing the integral has no poles, it can be expressed in the 
form 

where the function H (z) is holomorphic everywhere within the 
fundamental circle. 

Hence we have Poincare's theorem* that the integrals of a 
linear differential equation with algebraic coefficients can be ex- 
pressed as uniform functions of an appropriately chosen variable. 
* Acta Math., t. IV, p. 311. 



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16!).] INTEGRALS OF LINEAR EQUATIOKS 519 

169, The characteristic property of these uniform functions 
can be obtaiued as follows. Takiug the equation in the form 

where it is supposed that the term (if any) which involved ".-^^i 

has been removed from the equation by the usual substitution 
(§ 152), we denote by 0,, 0^, .... 8,, a fundamental system of 
integrals in the vicinity of any singularity (a^, i„). Let a closed 
path on the Eiemann surface, associated with the permanent 
equation, be described round the singularity; then, when the 
path is completed, the members of the fundamental system 
have acquired values ^i'. 6^, ..,, 6q, such that 

e^ = a["^l e, + al'^le^ + ... + a^^^l e^. {« = i, 2, . . . , 7), 

where the coefficients a*^' are constants such that their determ- 
inant is unity, because the derivative of order next to the highest 
is absent from the differential equation. 

Now ce and y are Fuchsian functions of z, existing only within 
the fundamental circle in the z-plane ; hence, when the path on 
the Riemann surface, which cannot be made evanescent, is com- 
pleted, ic and y return to their initial values, and z has described 
some path which is not evanescent. It follows, from the nature 
of the functions, that the end of the s-path is a point in another 
polygon, homologous with the initial position, so that the final 
position of z is of the form 

a^e + Q ^ 

7.^ + K ■ 

The integrals dt, d„ ..., 6q are uniform functions of z\ let them 
be denoted by 4>i{z), <fh(^)i •■■, 4'q(^)- Moreover, B„' is the value 
of Sa «t tJie conclusion of the path ; thus 



"Wz + bJ' 



so that the integrals in the fundamental system consist of a set of 
uniform functions of z, which are characterised by the property 



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520 ZETAFUCHSIA?J [169. 

Corresponding to the substitution of the Fuehsian group, we have 
a linear substitution S^ in the quantities 0i, 0^1 ■■-. 'f'g- the 
aggregate of these linear substitutious 8^, forms a group, which 
is isomorphic with the Fuchsian group. 

Functions of this pseudo-automorphic character are called* 
Zetafuchsian by Poincar^ : and thus we can say that linear differ- 
ential equations can be integrated by means of Fuchsian and Zeta- 
^cksian functions which are uniform. It is, however, necessary 
to obtain explicit expressions for the functions 0, in order that 
the equation may be regarded as integrated, This is effected 
(I.e.) by Poiocar^ as follows. 

Let 



represent the substitution inverse to yS^, so that the quantities 
A^^ are the minors of the determinant of S^. Take any q 
arbitrary rational functions of z, say Hi{s), S^i^), -.., Hqip); and 
by means of them, in association with the Fuchsian group, con- 
struct p infinite series, defined by the equations 



?,(.) = siA'''^ir.( 



s + hii'^i^+^if"" 



for the q values 1, ...,qoi fi.; the quantity m is a positive integer; 
and the summation with regard to i is over all the substitutions 
of the Fuchsian group. This integer ni is at our disposal : by 
choosing it sufficiently large, and by limiting the rational func- 
tions H, so that no one of the quantities 

is infinite on the fundamental circle, all the series can bo made 
absolutely converging: but we do not stay to establish this 
resulff. Assuming this convergence, and writing 

* Acta Math., t. v, p. 237. 

t It can be establishea on the same liaes as the convergence of Poincai^'s 
Thetafuchsian series: T. F., gg 304, 305. 



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169.] FUNCTIONS 521 

so that, for any value of k and al! the vahies of i, we get all the 
values of p for the group, we have 

But 

Owing to the properties of the isomorphic groups, we have 
and therefore 

SkS,'' = sr. 

that is, 

and therefore 

V7j;3 + 6i/ n=I l^" 

Now let 0(2) represent a Thetafuchsian series*, with the 
parametric integer m, and possessing the foregoing Kuchsian 
group: then, for each substitution of the group, we have 



^(S^')^(«^+«~^w- 



We introduce functions Zi, Z^, ..., Z^, defined by the relations 



(f = l ?)■ 



They satisfy the conditions 



and therefore we may take 

or the q functions Z, which are Zetafuchaian functions, constitute 
a system of integrals of the differential equation. 

170. As regards the Zetafuchsian functions thus constructed, 
it will be noted that the rational functions Hu ,.., Hq, which 



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522 PR0PBBTIK8 OF A [170. 

enter into their construction, are arbitrary; so that an infinite 
number of Zetafuchsian functions can be formed, admitting a 
Fucbsian group 6 and the linear group (say G) isomorphic with G. 
Further, the Thetafuchsiaa series @(z) with the parametric 
integer m is any whatever; but, as 

we have 

f^ _ f V \-f' l'^^l±_§A __ ^ 

so that we may take 

where P (cc, y) is any uniform function of le and y. The simplest 
case occurs when P{ie,y) = \. 
Again, we have 



z C"^ "^ 


A\ 


"ZM')* 


««2, 


W+...-K 


'*'/.<^)i 


"1?*^ + 


sj" 


and therefore 












1 




dZ. 


<!§-■ 




so that 
t. IV + /3A V7,. 


iT^:)=«^ 


dZ, 

■0 i 




d^S 

■■-C^ 

i 


that is. 












-zj"-^ 


z + 13. 


:)=cf 


+ "Z 




{41 rf^, 


da; \.7( 


,z + S, 


v. A, • 



are a Zetafuchsian system, admitting the Fuchsian group G and 
the isomorphic lineai- group G. 

The same property is possessed for all the derivatives of any 
order of the system Zi, Z,^, ..., Z^ with regard to x. 



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170,] ZETAFUGHSIAN SYSTEM 523 

Tliis property is used by Poincare to obtain the most general 
expression for a Zetafiichsian system, admitting the groups G and 
Q. Let it be T^, T,, ..., T^; and construct the matrix 



, dZ, 


dt-'Z, 


T, 


^" *.■■ 


■' d^^ ' 


, dZ, 


d'-'Z, 


T, 




d'-'Z, 
■' d^' ' 


T, 



Denote by (— l)°'~'A,i_] the determinant obtained by cutting out 
the a'" column from the matrix : then, by a known property of 
determinants, we have 



AA + A.!|- + ,.. + A,_ 


S^-.''.- 


ues of ». Hence 




„ a. . A, iZ, 
■'"^"a, " A, ■& 


A,_, di-^Z^ 



When s is subjected to any transformation of .the group 0, 
the quantities in any column in the matrix are subjected to the 
corresponding linear transformation of the group G ; so that each 
of the 5 + 1 determinants Ao, Aj, .... A^ is multiplied by the 
determinant of the linear transformation. Hence A, -h A^ is un- 
altered, that is, it is automorphic for the substitution of the 
group G\ and therefore, as this property is possessed for each 
substitution. A,- -;- A^ is automorphic for the group Q. Conse- 
quently, Ar -!- A, is a rational function of x and y, say 



?=-^.-. 



(f-O, 1, . ..,?-!)! 



T,..FA + F,-- 



dx 



. + n 






for 11 = 1, 2, ..., q. This is Poincar^'s expression for the most 
general Zetafuchsian system, admitting the Fuchsian group G 
and the isomorphic linear group G. 



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524 CONCLUDING [170. 

Wo can immediately verify that Z^, ..., Zg satisfy a linear 
differential equation, having coefficients that are rational in x 
and y. For 

d-iZ d^Z^ d'iZ^ 

da^ ' dx'i'' '"' da:^ ' 

are a Zetafuchsian system, admitting the Fuchsian group G and 
the isomorphic linear group G ; and therefore rational functions 
t^o, 1^, ..., i^j_i exist, such that 



holding for ail values of n. Thus Z^, ..., Zq are integrals of the 
linear differential equation 

Similarly, T„ ..., Tq are integra!s of a linear differential equation 
also of order q, having rational functions of x and y for its co- 
efficients, and characterised by the same groups G and G as 
characterise the equation satisfied hy Zi, ...,Zq. 



Concluding Remarks. 

171, The Zetafuchsian and Thetafuchsian functions thus 
used occur, for the most part, in the form of series of a particular 
kind; as they vpere first devised by Poincar^, his name is fre- 
quently associated with them. The main aim in constructing 
them was to obtain functions which should exhibit, simply and 
clearly, the organic character of automorphism under the substi- 
tutions of the groups; and they are avowedly intended* to be 
distinct in nature from series adapted to numerical calculation, 
such as series in powers of z. 

Unless both these properties, viz. the exhibition of the organic 
chai'acter of the function and its adaptability to numerical calcu- 
lation, are possessed by the functions involved, it is manifest that 
they are not in the most useful form. It is unlikely that the 
best development of the general theory can be effected, until 
" Acta Math., t. v, p. 211. 



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171.] REMARKS 625 

functions have been obtained in a form that possesses both the 
properties indicateci. In this connection, Klein* quotes a parallel 
instance from the theory of elliptic functions, viz. the series of the 
form 

S S (mw + wi'ft)')"", 

usedf by Eisenstein, which exhibit the characteristic antomorphic 
property of the modular functions, but are not adapted to nu- 
merical calculation Their deficiency in this respect has been 
met by the po&seBsion of the theta-functions and the sigma- 
functions The generalisation of the Jacobian th eta-function 
and the Weierstrassian sigma- function, required for automorphic 
functionb, has not yet been attained. 

We thus letum to the statement made at the beginning of 
the foiegoing sketch of Poincar^'s theory of linear differential 
equations with algebraic coefficients. The explicit analysis con- 
nected with the theory of automorphic functions has not yet 
acquired sufficiently comprehensive forms upon which to work ; 
and therefore its application to linear differential equations, as to 
any other subject, can be only partial and imperfect in its present 
st^;e. The theory of automorphic functions in general presents 
great possibilities of research : the gradual realisation of these 
possibilities will be followed by corresponding developments in 
many regions of analysis. 

• Vorlesungen S. lineare Differentialgleiehungen d. sweiten Ordnting, (Gottingen, 
1894}, p. 496. See also Fricke imd Klein, Theorie der automoTpken Functionen, 
t. II, p. 156. 

+ For references, see T.F.,% 56. 



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INDEX TO PART III. 





I. er 
ed 


g 


2'i 


' 


M- 




ed 


bl 


54 











t tl 

257. 






gul 






Algebraic coeMeieDts, ei^uatiotis having, 
Chapter x; oharnoter of integrals of, 
in vioinity of branch-point, 479, and 
in vicinity of a singularity, 480 ; mode 
of constructing integrals of, 483 ; 
Appell's class of, 484 ; aseocisited with 
automorphio fnnetiona, 488 (see auto- 
morphic fmictioas). 

Algebraio equation, roots of, satisfy a 
linear equation with rational ooefii- 
oienta, 46, 174 ; connected with differ- 
ential resolvents, 49. 

Algebraic integrals, eq.uations having, 
45, 165, Chapter v ; oonneoted with 
theory of finite groups, 175 ; connected 
with theory of oovariants, 175; equa- 
tions of second order having, 176 et 
Beq. ; equations of thin] order having, 
191 et seq. ; eqaationa of fourth order 
having, 201; constraction of, 184, 
198 ; and homogeneons forms, 202. 

Aji^ytieid form of group of integrals 
associated with multiple root of funda- 
mental equation of a singularity, 66 ; 
likewise for multiple root of funda- 
mental equation for a period or periods, 
416, 454. 

Anuulus, integral converging in any (see 
fundamental equation, irregular inte- 
gral, Lawent seriet). 

tmormaJes, 270. 

Apparent singularity, 117; conilitLons 
for, 119. 

Appell, 209, 484. 



tl I 117 

A t m pb t t d d ft t al 

equationshavingalgebiaic coefficients, 
48S ; and confonnal representation, 
491 ; associated with linear equations 
of second order, 495 ; constructed for 
a speoid case, 600 ; when there is one 
singularity, 6iD6; when there are two 
singularities, 508 ; when there are 
three, 509, 610; in general, 510 et 

Barnes, 448. 

begleitender bilinearer Differentialaua- 

dritek, 254. 
Benoit, 474. 
Bessel'E equation, 1, 13, 84, 100, 101, 

126, 164, 330, 333, 393. 
Bilinear ooDcomitant of two reciprocally 

adjoint equations, 254. 
Bdoher, 161, 169. 
B6tlier's theorem on equations of Fuchs- 

ian type with five singularities, 161. 
Boole, 229. 

Boulanger, 195, 197, 198. 
BrioscM, 206, a08, 218. 

Casorati, 55, 60, 417. 

Caochy, 11, 30. 

Canohy'a theorem used to establish 

existence of syneotic integral of a 

linear eq nation, 11. 
Cayley, 94, 113, 121, 182, 216, 233, 246, 



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528 



INDEX TO PAET 111 



Cels, 2S4. 

CharaoleriBtiG equation belonging to a 

singalurity, 40. 
Churacteristic equation toe deteimming 

fitctor of normal integrals, 294 ; effect 

of simple root of, 294; effect o£ 

multiple root, 398. 
Cbaracterietic function of an equation, 

Cbaracteristic index, definerl, 221 ; 

and namlier of regulav integrals 

of an equation, 930, 233 ; 
of reciprooally adjoint equatione, 
the eame, 257. 

ChrjBtal, 7, 83. 

Circular cjlinder, differential equation 
of, 164 ; (see BeseeVi equation). 

Class, equations of Fuchsian (see Ji'tic/is- 
ian type). 

Cockle, 49. 

CoefHcients, form of, near a eingularit; 
if all integtalB there are regular, 78. 

Collet, 20. 

Oonformal representation, and auto- 
morphic functions, 491 ; ami funda- 
mental polygon, 493. 

Constant ooefBoients, equation having, 
14—20. 

Construction, of regular integrals, liy 
method of Frobeniua, 78; of normal 
integrals, periodio integrals (see normal 
integrals, simply-periodie iniegraU, 
doubly-penodic integraU). 

Continnation prooeae applied to ftyneetio 
integral, 20. 

Continaed fractions used to obtain a 
fundamental equation, 439. 

Covariants associated viitli algebraic 
integrals, 202 ; for equations of third 
order, 203, 209; for equations of 
fourth order, 204 ; for equations of 
second order, 206. 

Craig, vi, 411. 

Crawford, 474. 

Curve, integral, defined, 303, HOS. 

Darbous, 20, 254, 475. 

Definite integrals (see Laplace's definite 

integral, double-loop integral). 
Determinant of a system of integrals, 
25 ; its value, 37 ; 

not vanishing, the system is fun- 
damental , 29 ; 
of a fundamentaJ system does not 
vanish, 30 ; 

il form of, for one particular 



syst 



, 34; 



form of, near a. aingalarity, 77 ; 
when the ooeffioienta are peri- 
odic, 406, 446 ; when the co- 
efB.cients are algebraic, 481. 
Determinants, infinite (see infinite de- 
tertainanta). 



Determining factor, of normal integral, 
262; obtained by Thome's meHiod, 
262 et seq.; conditions for, 265; tor 
' itegrals of Hamburger's equations, 



Diagonal of infinite determinant, 349. 
Difference-relations, 63, 417. 
Diffei-ential invariants (see invariants, 

differential). 
Differential resolvents, 49. 
Dini, 254, 256. 
Divisors, elementary (see elementary 






Double-loop integrals, 

integi'ate equations, mi ei seq. 

Doubly-periodic ooefBoienta, eqnaiiona 
having, 441 et seq. ; substitutions for 
the periods, 413 ; ^ndamental equa- 
tions for the periods, 444, 445. 

Doubly-periodio integrals of second kind, 
447 ; Ficaid's theorem on, 447 ; num- 
ber of, 448, 450 ; belonging to Lamp's 
equation, 463; how constructed, 471, 
475. 



applied t 



Blement of fundamental system. 30. 
Elementary divisors, of certain determ- 
inants, 41 — 43 ; of the fundamental 
equation, 65; determine groups and 
sub-groups of integrals. 62 : 

effect of, upon number of periodio 
integrals when coefficients are 
periodio, 416, 460. 
Elliott, M„ 434, 425, 474. 
Elliptic cylinder, differential equation 

of, 164, 399, 431—441. 
Expansion of converging infinite de- 
terminants, 363. 
Expansions, asymptotic (see asymptotic 

expamioiiK). 
Exponent, to which regular integral 
belongs, 74; properties of, 75; 

to which the det«rminant of a 
fundamental system belongs, 
77; 
to which normal integral belongs, 

262; 
of irregular integral as zero of an 
infinite determinant, 368, 
Exponents, snm of, for equations of 
Fuchsian type, 128 ; 

for Bieraann'a P-fnnotion, 139. 

Fabry, 94, 270. 

Factor, determining (see determining 
Jaetoi-). 

Fano, 214, 218. 

Finite groups of lineo-linear substitu- 
tions, in one variable, 176; connected 
with polyhedral functions, 181 ; asso- 
ciated with equations of second order 
having algebraic int^rals, 182 ; used 



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INDEX TO PART III 



529 



for con struct ion of algebra 
185; 



Integra 



J vftriablea, 193 ; their dif- 
ferential invariants, 195; the 
Laguerre invariaot, 196 ; used 
to construct equations of third 
order having algebraic integrals, 
197; 
in three variables, 300. 
First kind, periodic function of, 410. 
Floqnet, 231. 334, 259, 411, 418. 
Frioke, 489. 515, 535. 
Frobenius, 78. 93, 109, 336, 231, 233, 

338, 247, 254, 257, 369. 
Frobffliius' metliod, for the construction 
of integrals (ail being regular), 79 et 
aeq. ; variation of, suggested bj Cajley 
for some cases, 114; applied to hyper- 
geometi^c equation for special cases, 
147; for the construction of integrals, 
when only some are regolar, 335 et 
seq.; used for construction of irregular 
integrals, 379 et seq. 
Fuchs, L., 10, 11. 60, 65, 66, 79, 93, 94, 
109, 110, 117, 133, 135. 126, 129, 15S, 
306, 208, 216, 399, 462. 
Fnchsian equations, 123, 495 et seq. ; in- 
dependent variable a uniform function 
of quotient of integrals of. 496—199 ; 
mode of determining coefficients in. 
501; used as subsidiary to linearequa- 
tions of any order, 515. 
Fnohsion functions, associated with 
linear equations of the second order, 
500, 502, 516 ; associated with linear 
equations of general order. 615, 617; 
in the expression of integrals as uni- 
form functions, 620. 
Fnchsian group, {see Fuchsiaii /miction, 

Zetafuchsiam fanetioii). 
Focbsian type, equations of. Chapter iv, 
pp. 133 et seq.; form of, 123; proper- 
ties of exponents, 126 ; 

when fully determined by singu- 
larities and exponents, 128; 
of second order with any number 

of singularities, 150; 
forms of, when fc is an ordinary 
point, 152; when as is a singu- 
larity, 155, 158; Klein's normal, 
158; 
Lamp's equation transformed so 

as to be of, 160; 
equations of. haviag live singu- 
larities, 161; Bdcher's theorem 



Fundamental equation, belonging to a 
singularity, is same for all fuuda- 
mental systems, 38—40; invariants 
of, 10; Poincar^'s theorem on, 40; 



properties of, connected with ele- 
mentary divisors, 41—43; 
fundamental system of integrals 
assnoiated with 50 whe i roots 
are simple 52 vhen a root is 
multiple 53 
loots of how lelated to roots of 
mdioial equation 94 
Fundamental equation when integrals 
are iriegular expressed as an infinite 
deteimmant i(89 

luite terms 392 
I methods of obtaining 399 
Fundamental equations for double pen 
ods 441 145 their form 447 loots 
of determmedoubly peiioiio integrals 
of the second Mud, 448 ; number of 
these integrals, 150 ; efCeot of multiple 
roots of, 451. 
Fundamental equation for simple period, 
406; is invariantive, 406; form of, 
407 ; integral associated with a simple 
root, 408; integrals associated with a 
multiple root, 108; analytical expres- 
sion of, 419. 
Fundamental equation when coefficients 
are algebraic, 492; relation to in dicial 
equation. 482. 
Fundamental polygon for automorphic 

functions, 490, 493, 500. 
Fnndamental system of in tegrals, defined. 
30; its deteiininant is not evanescent, 
30; properties of , 30, 31 ; tests for,31, 
32; form of, near singularity, 50; if 
root of fundamental equation is sim- 
ple, 62; if root is multiple, 53; 

affected by elementary divisors 
of fundamental equation, 67; 
aggregate of groups associated 
with roots of indicia! equation 
make fundamental system. 95. 
Fundamental system, of irregular inte- 
grals, 387; of integrals when coelfioi- 
ents are simply-periodic, 408, 419; 
when ooeflioients are doubly-jwriodio, 
449 — 457; when coefGeients are alge- 
braic, 180. 
Fundamental system, constituted by 
group of integrals belonging to a 
multiple root of fundamental equa- 
tion (see grou,^ of integraU). 

Gordan, 183. 

Grade of normal integral, 269. 

Graf, 333. 

GreenhUl, 466. 

Glroup of tg 1 as led with mul- 
tiple t f f 1 tal equation, 
53 ; r sol d t b g oups, by ele- 
mentary d 57 Hamborger's 
sub-g p f 62 g al analytical 
form t 5 1 fundamental 
systen tit f 1 'er order, 72. 

34 



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INDEX TO PAET III 



Group of iotegralB, associated with mul- 
tiple root of iudioial eqaation in methocl 
of Frobeaiae, 80; general theorem on, 
93; aggregate of , inal^e a fundamentSil 
aystem, 96 ; compared with Ham- 
burger's groups, 113. 

Group of integrals for hjpergeometrio 
equation, 144. 

Group of irregular integrals associated 
with multiple root of characteristic 
infinite determinant. 381 et seq.; re- 
solved into sub-groups, 382. 

Group of integrals associated with mul- 
tiple roots of fundamental equationsfoi' 
periods when ooefficlents are doubly- 
periodic, 451 ; analytical expression of, 
482, 467; fnriiier development of, when 
uniform, 459. 

Group of integrals associated with mul- 
tiple root of fundamental equation for 
period when coefGcients are simply- 
periodic, 415 ; arranged in sub-groups, 
according to elementary divisors, 416; 
analytioi expression of, 419; they 
constitute a fondamental system for 
equation of lower order, 420 ; further 
expression of, when uniform, 421. 

Groups of substitutions, finite (see^mi(e 
groups); 

infinite (see automorphic ftinc- 

Griinfeba, 359. 
Gubler, 333. 
Gtinthev, 11, 299, 399. 
GjldSu, 469. 

Halphen, 254, 356, 281, 316, 316, 448, 

464, 465, 473. 
Hamburger, 38, 60, 62, 63, 64, 113, 977, 

380, 383, 286, 399, 489. 
Hamburger's equations, 276 et seq. ; 
of second order with normal ini 
grals, 979; the number of nc 
mai integrals, 280; 
ot general order n witii normal 
subnormal integrals, 288 et seq 
of third order with normal or sod- 
normal integrals, 301 et seq. 
Hamburger's sub-groups of integrals (seo 

sub-groups of integrals j. 
Hankel, 103, 333. 
Harley, 49. 
Heffter, 56, 156. 
Heine, 164, 166, 431, 441. 
Hermits, 15, 30, 448, 463, 465, 468, 473. 
Hermite, on equation with constant 
ooefUcienta, 15 — 20 ; on equation with 
doubly-periodic ooeffioients, 465. 



50. 
Hiil, G. W., 348, 38 
Hobson, 334, 337. 
Homogeneous forms {i 



398, 399, 403, 482. 



linear equations, defined, 
3; discussion limited to, 3. 

Homogeneous relations between inte- 
grals when they are algebraic, 203, 
217; of second degree for equations of 
third order, 230; and of higher degree, 
214. 

Horn, 333, 341, 342, 346, 347. 

Humbert, 167. 

Hypergeometrio function, used to render 
integrals of differential equations uni- 
form in speoial ease, 509. 

Hypergeometrio aeries, equation of, 1, 13, 
34, 103. 136, 144—160, 173, 338, 601, 
509. 

Identical relations, polynomial in powers 

of a logarithm, cannot exist, 69. 
Index, characteristic (see efiaraclei-UUe 
index) ; to which regular integral be- 
longs, 74; properties of, 75. 
Indiojal eqaation, when all integrals are 
regular, 85, 94; significance of, in the 
method of Frobeuius, 85 ; 

integral associated with a simple 

root of, 86; 
group of integrals associated with 

a multiple root of, 86; 
roots ot, how connected with roots 
of fundamental equation, 94; 
for equ t on w th not all integrals 
reg a 223 2''7 



ludioial f n t 
regular J4 

wh n not al ntegrals are regular, 

227, 
degree of, as affecting the number 
of regular intf^Is, 230, 933 ; 
of adjoint equation, as affecting 
the number of regular integrals, 
959. 
Infinite determinant, giving exponent of 
irregular integral, 368; modified to 
another determinant, 369; is a peri- 
odic function of its parameter, 375 ; 
effect of simple root of, 380. of a mul- 
tiple root of, 381 et seq. ; leads to the 
fundamental equation of the singu- 
larity, 389 ; expressed in finite terms, 
392. 
Infinite determinants in general, 349; 
convergence of, 360 ; properties of con- 
verging, in general, 353 et seq.; uni- 
form convergence of, wben functions 
of a parameter, 358 ; may be capa- 
ble of differentiation, 359; used to 
solve an unlimited number of linear 
equations, 360 ; applied to construct 
irregular integrals of differential equa- 



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INDEX TO PART III 



531 



Initial oondJtiouB defined, 4 

values, 4 ; efCeol of, apon form of 
synectic integral, 9 

Integral curve, 203, 20S, 

lot^rals. irregnkr (see i g la ate 
grali). 

Integrals, doubly-periodic irregular 
normal, regular, simply-pe o!o b 
norma], ayneotio (see unde these t ties 
respeetivdy). 

Intep^ rendered uuifona f net ons of 
a variable, wben there s one singu 
tarily, oOG ; when there are tno s Dgu 
laritias, 608 ; when there are th ee 
509, SIO; in general, 510 t sei by 
1 of Z tat oh i n f ct ona 518 



520 



I 



195 



f 



I 1 'f h 

quat f 

i 

f tl thi d 

fth f th 



01 213 
Laguerre'e, 196. 
Invariants of fundamental equation, oon- 
neoted with singnlarity, 38, 40; 
for irregular integrals, 398; 
connected with a period or periods 
405 i45 
h fli t Ig b 



il bl 



p th 

q t 
th d f 



Lab ange 251 

I ague e 196 

Lan^ eiuaton 1 12b 151 IbO 16o 
168 338 148 4b4— 473 

IJam^ a generahsed equation IbO 

Laplace a defin te ntegral aat sf Tng 
equat an vith ational ooefhc enta 
318 oonto r of 23 developed nto 
'.B here liie e exist 



324 e 



eq 



isp eas ng an reg lar 
ntegral 364 proof of convergence 
w thin BJ1 annnlne 3b6 

I/egeudre s equation 1 13 34 lOi 1 6 
IbO 163 

Liapounofl, 319, 42S — 431. 

Liapounoft'a theorem, applied to evaluate 
Laplace's definite integral, 324; me- 
thod of discussing uniform periodic 
integrals, 425. 

Lindemann, 4iU, 434, 437. 

Lindstedt, 439. 

Linear algebraic equations, infinite 
Eyetem of, solved by meana of infinite 
determinants, 360. 

Linear difierential equs^tion, definition 
of, a. 

Lineo-linear substitutiona (aee finite 



L g 






, quantity affected by, can 

uniform linear differential 

t and determine ita funda- 

tal atem, 66; 

d tical relatione, polynomial in 

fe lac iniegj^s free from, 106 j 

CO dition that some regular 

egral shall be free from, 



tai ed by g 1 sat t F b 




m th 1 379 th n.t t t Id 


M d 11 333. 


t 1 J tem 387 


M k ft 19. 






bl bym'^ ft 


M f nfinite determinants. 354. 


ph f t (ae in. pi 


M tt g Leffler, 399, 463. 


f ) 


M d 1 f notion, naed to reader inte- 




8 1 f differential eqaationa uniform 


da 197 200 Sii 334 33 341 


p al case, 610; Eiaenatain'a 


g e 113 


f aimUac to, 525. 




M It 1 1 oot, group of integrala asso- 


I in 150 153 15 15 161 17b 1 S 


tel th (see mi.itiyie root). 


1S7 190 117 tb 4 9 515 2 


M Itipli f periodic integral of second 


1 m I f rm f eq t f 


ki d 410; ia ft root of the funda- 


d d dF h typ 158 


m tal quation of the period, 406. 


m thod f eq ti f sec d 


M th 4 


d h m Ig b tefe 1 




176 


N m 1 f m, (after Frobenius) of equa- 


341 


t h ing some inlegrala regular. 


K h d4 35J 398 3')9 4 3 


27 1 component factors of such 


mm 14b 


eq tion, and of a composite 



( Iter Kieia) of equation of Fuohs- 



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